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STATiSTicS And probAbiliTy Topic 13 N LY Probability 13.1 Overview N EV What do you know? AL U AT IO Probability lies at the heart of nature....
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STATiSTicS And probAbiliTy

Topic 13

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Probability 13.1 Overview

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What do you know?

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Probability lies at the heart of nature. Think about all the events that had to happen for you to be born, for example . . . the odds are extraordinary. Probability is that part of mathematics that gives meaning to the idea of uncertainty, of not fully knowing or understanding the occurrence of some event. We often hear that there is a good chance of rain, people bet with different odds that a favourite horse will win at Caulfield, and so on. In each case, we are making a guess as to what will be the outcome of some event. It is important to learn about probability so that you can understand that chance is involved in many decisions that you will have to make in your life and in everyday events.

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Why learn this?

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1 THinK List what you know about probability. Use a thinking tool such as a concept map to show your list. 2 pAir Share what you know with a partner and then with a small group. 3 SHArE As a class, create a thinking tool such as a large concept map to show your class’s knowledge about probability.

Learning sequence

Overview Theoretical probability Experimental probability Venn diagrams and two-way tables Two-step experiments Mutually exclusive and independent events Conditional probability Review ONLINE ONLY

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13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8

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WATcH THiS vidEo The story of mathematics: What are the chances? Searchlight id: eles-1700

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13.2 Theoretical probability The language of probability

Unlikely

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• The probability of an event is a measure of the likelihood that the event will take place. • If an event is certain to occur, then it has a probability of 1. • If an event is impossible, then it has a probability of 0. • The probability of any other event taking place is given by a number between 0 and 1. • The higher the probability, the more likely it is for the event to occur. • Descriptive words such as ‘impossible’, ‘unlikely’, ‘likely’ and ‘certain’ are commonly used when referring to the chance of an event occurring. Some of these are shown on the probability scale below.

Even chance

Likely

0.25

0.5

0.75

1

50%

WorKEd EXAmplE 1

100%

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0

Certain

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Impossible

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On the probability scale given at right, insert each 0 of the following events at appropriate points. a You will sleep tonight. b You will come to school the next Monday during a school term. c It will snow in Victoria this year. THinK

Carefully read the given statement and label its position on the probability scale.

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b

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Provide reasoning.

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Carefully read the given statement and label its position on the probability scale.

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Provide reasoning.

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a

0.5

a

0

0.5

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Under normal circumstances, I will certainly sleep tonight. b

b

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0.5

1

It is very likely but not certain that I will come to school on a Monday during term. Circumstances such as illness or public holidays may prevent me from coming to school on a specific Monday during a school term.

Maths Quest 9

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Carefully read the given statement and label its position on the probability scale.

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c c (Summer)

0

Provide reasoning.

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c (Winter)

0.5

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It is highly likely but not certain that it will snow in Victoria during winter. The chance of snow falling in Victoria in summer is highly unlikely but not impossible.

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c

Key terms

WorKEd EXAmplE 2

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• The study of probability uses many special terms that must be clearly understood. Here is an explanation of some of the more common terms. Chance experiment: A chance experiment is a process, such as rolling a die, that can be repeated many times. Trial: A trial is one performance of an experiment to get a result. For example, each roll of the die is called a trial. Outcome: The outcome is the result obtained when the experiment is conducted. For example, when a normal six-sided die is rolled the outcome can be 1, 2, 3, 4, 5 or 6. Sample space: The set of all possible outcomes is called the sample space and is given the symbol ξ. For the example of rolling a die, ξ = {1, 2, 3, 4, 5, 6}. Event: An event is the favourable outcome of a trial and is often represented by a capital letter. For example, when a die is rolled, A could be the event of getting an even number; A = {2, 4, 6}. Favourable outcome: A favourable outcome for an event is any outcome that belongs to the event. For event A above (rolling an even number), the favourable outcomes are 2, 4 and 6.

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For the chance experiment of rolling a die: a list the sample space b list the events: i rolling a 4 ii rolling an even number iii rolling at least 5 iv rolling at most 2 c list the favourable outcomes for: i {4, 5, 6} ii not rolling 5 iii rolling 3 or 4 iv rolling 3 and 4. THinK a b

WriTE

The outcomes are the numbers 1 to 6.

a

This describes only 1 outcome.

b

i ii

The possible even numbers are 2, 4 and 6.

ξ = {1, 2, 3, 4, 5, 6} i ii

{4} {2, 4, 6}

Topic 13 • Probability 473

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iii

‘At least 5’ means 5 is the smallest.

iii

{5, 6}

iv

‘At most 2’ means 2 is the largest.

iv

{1, 2}

i

The outcomes are shown inside the brackets.

i

4, 5, 6

ii

‘Not 5’ means everything except 5.

c

c

ii

1, 2, 3, 4, 6

The event is {3, 4}.

iii

3, 4

iv

There is no number that is both 3 and 4.

iv

There are no favourable outcomes.

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Theoretical probability

P(Tails) = 12.

and

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• When a coin is tossed, there are two possible outcomes, Heads or Tails. That is, ξ = {H, T}. • In ideal circumstances, the two outcomes have the same likelihood of occurring, so they are allocated the same probability. For example, P(Heads) = 12 (This says the probability of Heads = 12.)

number of favourable outcomes . total number of outcomes

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P(A) =

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• The total of the probabilities equals 1, as there are no other possible outcomes. • In general, if all outcomes are equally likely to occur (ideal circumstances), then the probability of event A occurring is given by

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WorKEd EXAmplE 3

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A die is rolled and the number uppermost is noted. Determine the probability of each of the following events. a A = {1} b B = {odd numbers} c C = {4 or 6} THinK

WriTE

There are 6 possible outcomes. a

A has 1 favourable outcome.

a

P(A) = 16

b

B has 3 favourable outcomes: 1, 3 and 5.

b

P(B) = 36

C has 2 favourable outcomes.

c

c

= 12 P(C) = 26 = 13

474

Maths Quest 9

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Exercise 13.2 Theoretical probability individUAl pATHWAyS ⬛

prAcTiSE



Questions: 1–16

conSolidATE



Questions: 1–18 ⬛ ⬛ ⬛ Individual pathway interactivity

rEFlEcTion Write a sentence using the word ‘probability’ that shows its meaning.

mASTEr

Questions: 1–19 int-4534

WE1 On the given probability scale, insert each of the 0 0.5 following events at appropriate points. a The school will have a lunch break on Friday. b Australia will have a swimming team in the Commonwealth Games. c Australia will host two consecutive Olympic Games. d At least one student in a particular class will obtain an A for Mathematics. e Mathematics will be taught in secondary schools. f In the future most cars will run without LPG or petrol. g Winter will be cold. h Bean seeds, when sown, will germinate. Indicate the chance of each event listed in question 1 using one of the following terms: certain, likely, unlikely, impossible. WE2a For each chance experiment below, list the sample space. a Rolling a die b Tossing a coin c Testing a light bulb to see whether it is defective or not d Choosing a card from a normal deck and noting its colour e Choosing a card from a normal deck and noting its suit WE2b A normal 6-sided die is rolled. List each of the following events. a Rolling a number less than or equal to 3 b Rolling an odd number c Rolling an even number or 1 d Not rolling a 1 or 2 e Rolling at most a 4 f Rolling at least a 5 WE2c A normal 6-sided die is rolled. List the favourable outcomes for each of the following events. a A = {3, 5} b B = {1, 2} c C = ‘rolling a number greater than 5’ d D = ‘not rolling a 3 or a 4’ e E = ‘rolling an odd number or a 2’ f F = ‘rolling an odd number and a 2’ g G = ‘rolling an odd number and a 3’

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Topic 13 • Probability 475

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STATistics and probability

A card is selected from a normal deck of 52 cards and its suit is noted. a List the sample space. b List each of the following events. i ‘Drawing a black card’ ii ‘Drawing a red card’ iii ‘Not drawing a heart’ iv ‘Drawing a black or a red card’ 7 How many outcomes are there for: a rolling a die b tossing a coin c drawing a card from a standard deck d drawing a card and noting its suit e noting the remainder when a number is divided by 5?

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6

A card is drawn at random from a standard deck of 52 cards. Note: ‘At random’ means that every card has the same chance of being selected. Find the probability of selecting: a an ace b a king c the 2 of spades d a diamond.   WE3  9 A card is drawn at random from a deck of 52. Find the probability of each event below. a A = {5 of clubs} b B = {black card} c C = {5 of clubs or queen of diamonds} d D = {hearts} 8

476  Maths Quest 9

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E = {hearts or clubs} f F = {hearts and 5} g G = {hearts or 5} h H = {aces or kings} i I = {aces and kings} j J = {not a 7} 10 A letter is chosen at random from the letters in the word PROBABILITY. What is the probability that the letter is: a B b not B c a vowel d not a vowel? 11 The following coloured spinner is spun and the colour is noted. What is the probability of each of the events given below? a A = {blue} b B = {yellow} c C = {yellow or red} d D = {yellow and red} e E = {not blue}

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UndErSTAndinG

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A bag contains 4 purple balls and 2 green balls. If a ball is drawn at random, then what is the probability that it will be: i purple ii green? b Design an experiment like the one in part a but where the probability of drawing a purple ball is 3 times that of drawing a green ball. 13 Design spinners (see question 11) using red, white and blue sections so that: a each colour has the same probability of being spun b red is twice as likely to be spun as either of the other 2 colours c red is twice as likely to be spun as white and 3 times as likely to be spun as blue.

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12 a

Do you think that the probability of tossing Heads is the same as the probability of tossing Tails if your friend tosses the coin? What are some reasons that it might not be? 15 If the following four probabilities were given to you, which two would you say were not correct? Give reasons why. 0.725, −0.5, 0.005, 1.05

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int-0089

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problEm SolvinG 16

Consider this spinner. Discuss whether the spinner has equal chance of falling on each of the colours.

Topic 13 • Probability 477

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STATistics and probability

A box contains two coins. One is a double-headed coin, and the other is a normal coin with Heads on one side and Tails on the other. You draw one of the coins from a box and look at one of the sides. It is Heads. What is the probability that the other side shows Heads also? 18 ‘Unders and Overs’ is a game played with two normal six-sided dice. The two dice are rolled, and the numbers uppermost added to give a total. Players bet on the outcome being ‘under 7’, ‘equal to 7’ or ‘over 7’. If you had to choose one of these outcomes, which would you choose? Explain why. 19 Justine and Mary have designed a new darts game for their Year 9 Fete Day. Instead of a circular dart board, their dart board is in the shape of two equilateral triangles. The inner triangle (bullseye) has a side length of 3 cm, while the outer triangle has side length 10 cm.

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10 cm

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3 cm

Given that a player’s dart falls in one of the triangles, what is the probability that it lands in the bullseye? Write your answer correct to 2 decimal places.

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13.3 Experimental probability Relative frequency

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•• A die is rolled 12 times and the outcomes are recorded in the table below. 1

2

3

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5

6

Frequency

3

1

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3

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Outcome

In this chance experiment there were 12 trials. The table shows that the number 1 was rolled 3 times out of 12.

3 = 14. •• So the relative frequency of 1 is 3 out of 12, or 12

As a decimal, the relative frequency of 1 is equal to 0.25. •• In general, the relative frequency of an outcome =

the frequency of the outcome . total number of trials

If the number of trials is very large, then the relative frequency of each outcome becomes very close to the theoretical probability.

478  Maths Quest 9

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WorKEd EXAmplE 4

For the chance experiment of rolling a die, the following outcomes were noted. Outcome

1

2

3

4

5

6

Frequency

3

1

4

6

3

3

THinK

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How many trials were there? How many threes were rolled? What was the relative frequency for each number written as decimals? WriTE

Adding the frequencies will give the number of trials.

a

1 + 3 + 4 + 6 + 3 + 3 = 20 trials

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The frequency of 3 is 4.

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4 threes were rolled.

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Add a relative frequency row to the table and complete it.

c

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Frequency

3 3 20

= 0.15

3

4

1

4

1 20

4 20

= 0.05

= 0.2

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6 20

= 0.3

5

6

3

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3 20

= 0.15

3 20

= 0.15

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Relative frequency

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Outcome

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a b c

Group experiment

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• Organise for the class to toss a coin at least 500 times. For example, if the class has 20 students, each one should record 25 outcomes and enter their information into a grid as shown below.

B C

Total T

Total H

Relative frequency (Tails)

17 8

17

8

25

17 25

= 0.68

8 25

= 0.32

15 10

32

18

50

32 50

= 0.64

18 50

= 0.36

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etc.

T

Relative frequency (Heads)

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H

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Group

Total outcomes

• Questions: 1. In an ideal situation, what would you expect the relative frequencies to be? Has this occurred? 2. As more information was added to the table, what happened to the relative frequencies? 3. What do you think might happen if the experiment was continued for another 500 tosses? • A rule called the law of large numbers indicates that as the number of trials increases, then the relative frequencies will tend to get closer to the expected value (in this case 0.5).

Topic 13 • Probability 479

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STATiSTicS And probAbiliTy

Experimental probability • Sometimes it is not possible to calculate theoretical probabilities and in such cases experiments, sometimes called simulations, are conducted to determine the experimental probability. • The relative frequency is equal to the experimental probability. Experimental probability =

the frequency of the outcome total number of trials

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For example, the spinner shown at right (made from light cardboard and a toothpick) is not symmetrical, and the probability of each outcome cannot be determined theoretically. However, the probability of each outcome can be found by using the spinner many times and recording the outcomes. If a large number of trials is conducted, the relative frequency of each outcome will be very close to its probability.

WorKEd EXAmplE 5

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The spinner shown above was spun 100 times and the following results were achieved. Outcome

1

2

3

4

Frequency

7

26

9

58

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How many trials were there? What is the experimental probability of each outcome? What is the sum of the 4 probabilities?

THinK

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a b c

WriTE

Adding the frequencies will determine the number of trials.

a

7 + 26 + 9 + 58 = 100 trials

b

The experimental probability equals the relative frequency.

b

7 P(1) = 100

= 0.07

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a

26 P(2) = 100

= 0.26

9 P(3) = 100

= 0.09 58 P(4) = 100

= 0.58 c

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Add the probabilities (they should equal 1).

c

0.07 + 0.26 + 0.09 + 0.58 = 1

Maths Quest 9

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STATiSTicS And probAbiliTy

Exercise 13.3 Experimental probability individUAl pATHWAyS ⬛

prAcTiSE



Questions: 1–4, 6–13, 19–21, 25

conSolidATE



Questions: 1–4, 6–11, 15, 16, 22–25 ⬛ ⬛ ⬛ Individual pathway interactivity

mASTEr

int-4535

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FlUEncy

Each of the two tables below contains the results of a chance experiment (rolling a die). For each table, find: i the number of trials held ii the number of fives rolled iii the relative frequency for each outcome, correct to 2 decimal places iv the sum of the relative frequencies. WE4

2

4

1

4

5

6

49

40

46

2

3

4

Frequency

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1

5

Number

1

2

3

Frequency

52

38

45

a

Outcome

H

Frequency

22

Outcome

H

T

Frequency

31

19

T

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b

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A coin is tossed in two chance experiments. The outcomes are recorded in the tables below. For each experiment, find: i the relative frequency of both outcomes ii the sum of the relative frequencies.

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rEFlEcTion What are the most important similarities between theoretical and experimental probability calculations?

Questions: 1–3, 5–8, 12–14, 16–25

Construct an irregular spinner using cardboard and a toothpick. By carrying out a number of trials, estimate the probability of each outcome. 4 WE5 An unbalanced die was rolled 200 times and the following outcomes were recorded.

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Number

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4

5

6

Frequency

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32

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73

Using these results, find: P(6) b P(odd number) c P(at most 2) d P(not 3). a

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5

A box of matches claims on its cover to contain 100 matches. A survey of 200 boxes established the following results. Number of matches

95

96

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99

100 101 102 103 104

Frequency

1

13

14

17

27

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If you were to purchase a box of these matches, what is the probability that: a the box would contain 100 matches b the box would contain at least 100 matches c the box would contain more than 100 matches d the box would contain no more than 100 matches?

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Here is a series of statements based on experimental probability. If a statement is not reasonable, give a reason why. a I tossed a coin 5 times and there were 4 Heads, so P(H) = 0.8. b Sydney Roosters have won 1064 matches out of the 2045 that they have played, so P(Sydney will win their next game) = 0.54. c P(The sun will rise tomorrow) = 1. d At a factory, a test of 10 000 light globes showed that 7 were faulty. Therefore, P(faulty light globe) = 0.0007. e In Sydney it rains an average of 143.7 days each year, so P(it will rain in Sydney on the 17th of next month) = 0.39. 7 At a birthday party, some cans of soft drink were put in a container of ice. There were 16 cans of Coke, 20 cans of Sprite, 13 cans of Fanta, 8 cans of Sunkist and 15 cans of Pepsi. If a can was picked at random, what is the probability that it was: a a can of Pepsi b not a can of Fanta? 8 MC In Tattslotto, 6 numbers are drawn from the numbers 1, 2, 3, . . . 45. The number of different combinations of 6 numbers is 8 145 060. If you buy 1 ticket, what is the probability that you will win the draw? 1 8 145 060

b

1 45

c

45 8 145 060

d

1 6

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6 8 145 060

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Maths Quest 9

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STATistics and probability

If a fair coin is tossed 400 times, how many Tails are expected? 10 If a fair die is rolled 120 times, how many threes are expected? 11   WE9    MC  A survey of high school students asked ‘Should Saturday be a normal school day?’ 350 students voted yes, and 450 voted no. What is the probability that a student chosen at random said no? 9   WE8 

A

7 9 7 B C 16 16 9

D

9 14

1 350

E

In a poll of 200 people, 110 supported party M, 60 supported party N and 30 were undecided. If a person is chosen at random from this group of people, what is the probability that he or she: a supports party M  b supports party N c supports a party  d is not sure what party to support?

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A random number is picked from N = {1, 2, 3, . . . 100}. What is the probability of picking a number that is: a a multiple of 3 b a multiple of 4 or 5 c a multiple of 5 and 6?

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The numbers 3, 5 and 6 are combined to form a three-digit number such that no digit may be repeated. a i  How many numbers can be formed? ii  List them. b Find P(the number is odd). c Find P(the number is even). d Find P(the number is a multiple of 5). 15   MC  In a batch of batteries, 2 out of every 10 in a large sample were faulty. At this rate, how many batteries are expected to be faulty in a batch of 1500? A 2 B 150 C 200 D 300 E 750 16 Svetlana, Sarah, Leonie and Trang are volleyball players. The probabilities that they will score a point on serve are 0.6, 0.4, 0.3 and 0.2 respectively. How many points on serve are expected from each player if they serve 10 times each?

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Topic 13 • Probability  483

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STATistics and probability

A survey of the favourite leisure activity of 200 Year 9 students produced the following results.

17   MC 

Activity Number of students

Playing sport Fishing Watching TV Video games Surfing 58 26 28 38 50

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The probability (given as a percentage) that a student selected at random from this group will have surfing as their favourite leisure activity is: A 50% B 100% C 25% D 0% E 29% 18 The numbers 1, 2 and 5 are combined to form a three-digit number, allowing for any digit to be repeated up to three times. a How many different numbers can be formed? b List the numbers. c Determine P(the number is even). d Determine P(the number is odd). e Determine P(the number is a multiple of 3).

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REASONING

John has a 12-sided die numbered 1 to 12 and Lisa has a 20-sided die numbered 1 to 20. They are playing a game where the first person to get the number 10 wins. They are rolling their dice individually. a Find P(John gets a 10). b Find P(Lisa gets a 10). c Is this game fair? Explain. 20 At a supermarket checkout, the scanners have temporarily broken down and the cashiers must enter in the bar codes manually. One particular cashier overcharged 7 of the last 10 customers she served by entering the incorrect bar code. a Based on the cashier’s record, what is the probability of making a mistake with the next customer? b Should another customer have any objections with being served by this cashier? c Justify your answer to part b. 21 If you flip a coin 6 times, how many of the possible outcomes could include a Tail on the second toss?

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problem solving

In a jar, there are 600 red balls, 400 green balls, and an unknown number of yellow balls. If the probability of selecting a green ball is 15, how many yellow balls are in the jar? 23 In another jar there are an unknown number of balls, N, with 20 of them green. The other colours contained in the jar are red, yellow and blue, with P(red or yellow) = 12, P(red or green) = 14 and P(blue) = 13. Determine the number of red, yellow and blue balls in the jar. 24 The gender of babies in a set of triplets is simulated by flipping 3 coins. If a coin lands Tails up, the baby is a boy. If a coin lands Heads up, the baby is a girl. In the simulation, the trial is repeated 40 times. The following results show the number of Heads obtained in each trial:  0, 3, 2, 1, 1, 0, 1, 2, 1, 0, 1, 0, 2, 0, 1, 0, 1, 2, 3, 2, 1, 3, 0, 2, 1, 2, 0, 3, 1, 3, 0, 1, 0, 1, 3, 2, 2, 1, 2, 1. a Calculate the probability that exactly one of the babies in a set of triplets is female. b Calculate the probability that more than one of the babies in the set of triplets is female.

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STATiSTicS And probAbiliTy

A survey of the favourite foods of Year 9 students is recorded, with the following results.

Hamburger

45

Fish and chips

31

Macaroni and cheese

30

Lamb souvlaki

25

BBQ pork ribs

21

Cornflakes

17

T-bone steak

14

Banana split

12 9

Hot dogs

8

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Corn-on-the-cob

Garden salad

8

Veggie burger

7

Fruit salad

6 5

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Smoked salmon Muesli

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Tally

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Estimate the probability that macaroni and cheese is the favourite food of a randomly selected Year 9 student. b Estimate the probability that a vegetarian dish is a randomly selected student’s favourite food. c Estimate the probability that a beef dish is a randomly selected student’s favourite food.

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13.4 Venn diagrams and two-way tables The complement of an event

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• Suppose that a die is rolled: ξ = {1, 2, 3, 4, 5, 6}. If A is the event ‘rolling an odd number’, then A = {1, 3, 5}. • There is another event called ‘the complement of A’, or ‘not A’. This event contains all the outcomes that do not belong to A. It is given the symbol A′. • In this case A′ = {2, 4, 6}. • A and A′ can be shown on a Venn diagram. ξ

ξ

A

1 3 5

4

6 2

A is coloured.

A

1 3 5

4

6 2

A′ (not A) is coloured. Topic 13 • Probability 485

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STATiSTicS And probAbiliTy

WorKEd EXAmplE 6

For the sample space ξ = {1, 2, 3, 4, 5}, list the complement of each of the following events. a A = {multiples of 3} b B = {square numbers} c C = {1, 2, 3, 5} WriTE

The only multiple of 3 in the set is 3. Therefore A = {3}. A′ is every other element of the set.

b

The only square numbers are 1 and 4. Therefore B = {1, 4}. B′ is every other element of the set.

c

C = {1, 2, 3, 5}. C′ is every other element of the set.

a

A′ = {1, 2, 4, 5}

b

B′ = {2, 3, 5}

O

a

N LY

THinK

C′ = {4}

N

c

AT IO

Venn diagrams and two-way tables Venn diagrams

ξ

A

ξ

2

ξ

A

B

3

7

8

B 2

A 3

7

B 2

7 8

8

M PL

E

3

EV

AL U

• Venn diagrams convey information in a concise manner and are often used to illustrate sample spaces and events. Here is an example. – In a class of 20 students, 5 study Art, 9 study Biology, and 2 students study both subjects. – This information is shown on the diagram below, where A = {students who study Art} and B = {students who study Biology}.

A contains 5 students.

B contains 9 students.

Note: In the case shown above, 8 students in the class study neither Art nor Biology.

SA

• The Venn diagram has 4 regions, each with its own name. A∩B ξ

A 3

ξ B

2

A 3

7 8

There are 2 students who study Art and Biology. They occupy the region called ‘A and B’ or A ∩ B.

486

A ∩ B′ B 2

7 8

There are 3 students who study Art but not Biology. They occupy the region called ‘A and not B’ or A ∩ B′.

Maths Quest 9

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STATiSTicS And probAbiliTy

ξ

A′ ∩ B A 3

A′ ∩ B′

ξ A

B 2

3

7

B 2

7

8

8

The remaining 8 students study neither subject. They occupy the region called ‘not A and not B’ or A′ ∩ B′.

N LY

There are 7 students who study Biology but not Art. They occupy the region called ‘not A and B’ or A′ ∩ B.

• The information can also be summarised in a two-way table. Total 5 15

9

11

20

Total

N

Not Biology 3 8

AT IO

Art Not Art

Biology 2 7

O

Two-way tables

Note: Nine students in total study Biology and 11 do not. Five students in total study Art and 15 do not.

AL U

Number of outcomes

• If event A contains 7 outcomes or members, this is written as n(A) = 7. • So n(A ∩ B′) = 3 means that the event ‘A and not B’ has 3 outcomes.

EV

WorKEd EXAmplE 7

THinK

Identify the regions showing M and add the outcomes.

M

6

11 15

4 WriTE/drAW a

SA

a

M PL

E

For the Venn diagram shown, write down the number of outcomes in each of the following. aM b M′ c M∩N ξ N d M ∩ N′ e M′ ∩ N′

ξ M 6

N 11 15 4

n(M) = 6 + 11 = 17 b

Identify the regions showing M′ and add the outcomes.

b

ξ M 6

N 11 15 4

n(M′) = 4 + 15 = 19

Topic 13 • Probability 487

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STATiSTicS And probAbiliTy

c

M ∩ N means ‘M and N’. Identify the region.

ξ

c

M 6

N 11 15 4

n(M ∩ N) = 11 M ∩ N′ means ‘M and not N’. Identify the region.

d

ξ M 6

N 11 15

N LY

d

4

n(M ∩ N′) = 6 M′ ∩ N′ means ‘not M and not N’. Identify the regions.

e

ξ

N 11 15

O

e

M

4

N

6

AT IO

n(M′ ∩ N′) = 4

AL U

WorKEd EXAmplE 8

EV

Show the information from the Venn diagram on a two-way table.

THinK

M PL SA

2

3

488

A 3

There are 7 elements in A and B. There are 3 elements in A and ‘not B’. There are 2 elements in ‘not A’ and B. There are 5 elements in ‘not A’ and ‘not B’.

B 7 2 5

WriTE

Draw a 2 × 2 table and add the labels A, A′, B and B′.

E

1

ξ

A

A′

A

A′

B

7

2

B′

3

5

B B′

Add in a column and a row to show the totals.

A

A′

Total

B

7

2

9

B′

3

5

8

Total

10

7

17

Maths Quest 9

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STATiSTicS And probAbiliTy

WorKEd EXAmplE 9

Left-handed

Right-handed

7

20

17

48

Blue eyes Not blue eyes

n(L ∩ B) = 7 n(L ∩ B′) = 17 n(L′ ∩ B) = 20 n(L′ ∩ B′) = 48

ξ L 17

7

B 20 48

N

2

Draw a Venn diagram that includes a sample space and events L for left-handedness and B for blue eyes. (Right-handedness = L′)

AT IO

1

drAW

O

THinK

N LY

Show the information from the two-way table on a Venn diagram.

Event A or B

M PL

E

EV

AL U

• This Venn diagram illustrates the results of a survey of 20 people, showing whether they drink tea and whether they drink coffee. In all there are 19 people who drink tea or coffee. They are found in the shaded region of the diagram. ξ This large group of people is written as C ∪ T and T called ‘C or T ’. Note that the people who drink both tea and coffee, C ∩ T, are included in this group. C 5 n(C ∪ T) = 19 12 A number of people drink tea or coffee, but not 2 both. This group contains the 2 people who drink only tea and the 5 people who drink only coffee. 1 n(people who drink tea or coffee, but not both) = 7 WorKEd EXAmplE 10

SA

In a class of 24 students, 11 students play basketball, 7 play tennis, and 4 play both sports. a Show the information on a Venn diagram. b If one student is selected at random, then find the probability that: i the student plays basketball ii the student plays tennis or basketball iii the student plays tennis or basketball but not both. THinK a

1

Draw a sample space with events B and T.

drAW a

ξ B

T

Topic 13 • Probability 489

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STATiSTicS And probAbiliTy

n(B ∩ T) = 4 n(B ∩ T′) = 11 − 4 = 7 n(T ∩ B′) = 7 − 4 = 3 So far, 14 students out of 24 have been placed. n(B′ ∩ T′) = 24 − 14 = 10

2

Identify the number of students who play basketball.

i

ξ B 7

B 7

T 4 3 10

b

i

number of students who play basketball total number of students n(B) = 24 11 = 24

P(B) =

N LY

b

ξ

T 4 3

number of favourable outcomes total number of outcomes

Identify the number of students who play tennis or basketball. ξ B 7

ii

T 4 3

AL U

10

Identify the number of students who play tennis or basketball but not both. ξ B 7

T 4 3

iii

=

M PL

E

10

n(B ∩ T′) + n(B′ ∩ T) = 3 + 7 = 10 P(tennis or basketball but not both) 10 24 5 = 12

EV

iii

n(T ∪ B) 24 14 = 24 7 = 12

P(T ∪ B) =

AT IO

ii

N

P B =

O

10

Exercise 13.4 Venn diagrams and two-way tables individUAl pATHWAyS ⬛

SA

rEFlEcTion How will you remember the difference between when one event and another occurs and when one event or another occurs?

prAcTiSE

Questions: 1–5, 7, 9, 11, 13, 15–17



conSolidATE



Questions: 1–4, 6, 8, 10, 13–15, 17–19 ⬛ ⬛ ⬛ Individual pathway interactivity

mASTEr

Questions: 1, 3, 4, 6, 8, 10, 12, 14, 16–20 int-4536

FlUEncy 1

490

If ξ = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, list the complement of each of the following events. a A = {multiples of 3} b B = {numbers less than 20} c C = {prime numbers} d D = {odd numbers and numbers greater than 16} WE6

Maths Quest 9

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STATiSTicS And probAbiliTy

For the Venn diagram shown, write down the number of outcomes in: a ξ b S c T d S∩T e T∩S f T ∩ S′ g S′ ∩ T′. 3 WE8 Show the information from question 2 on a two-way table. 4 WE9 Show the information from this two-way table on a Venn diagram. W′

V

21

7

V′

2

10

S 5

6

T 7 9

doc-6311

N LY

W

ξ

doc-6312

O

For each of the following Venn diagrams, use set notation to write the name of the region coloured in: i purple ii pink. a

b

ξ W

ξ

c A

B

ξ

N

5

WE7

AT IO

2

A

B

The membership of a tennis club consists of 55 men and 45 women. There are 27 left-handed people, including 15 men. a Show the information on a two-way table. b Show the information on a Venn diagram. c If one member is chosen at random, find the probability that the person is: i right-handed ii a right-handed man iii a left-handed woman. 7 Using the information given in the Venn diagram, if one outcome is chosen at random, find: a P(L) b P(L′) c P(L ∩ M) d P(L ∩ M′). WE10 8 Using the information given in the table, if one family is chosen at random, find the probability that they own:

SA

M PL

E

EV

AL U

6

ξ L 3

5

M 7 10

Pets owned by families Cat

No cat

Dog

4

11

No dog

16

9

a cat a cat and a dog c a cat or a dog or both d a cat or a dog but not both e neither a cat nor a dog. a

b

Topic 13 • Probability 491

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STATistics and probability

ξ group of athletes was surveyed and the results were shown on S a Venn diagram. L 3 5 S = {sprinters} and L = {long jumpers}. 2 a How many athletes were included in the survey? b If one of the athletes is chosen at random, what is the probability that the athlete competes in: i long jump ii long jump and sprints iii long jump or sprints iv long jump or sprints but not both?

9 A

N LY

UNDERSTANDING

6

If ξ = {children}, S = {swimmers} and R = {runners}, describe in words each of the following. a S′ b S ∩ R c R′ ∩ S′ d R ∪ S 11 A group of 12 students was asked C H whether they liked hip hop (H) and Ali ✓ ✓ whether they liked classical music Anu (C). The results are shown in the Chris ✓ table below. a Show the results on: George ✓ i             a Venn diagram Imogen ✓ i       i a two-way table. Jen ✓ ✓ b If one student is selected at Luke ✓ ✓ random, find: Pam ✓ i             P(H) Petra i        i P(H ∪ C) Roger ✓ i   ii P(H ∩ C) Seedevi ✓ iv  P(student likes classical or hip Tomas hop but not both).

SA

M PL

E

EV

AL U

AT IO

N

O

10

12

Place the elements of the following sets of numbers in their correct position in a single Venn diagram. A = {prime numbers from 1 to 20} B = {even numbers from 1 to 20} C = {multiples of 5 from 1 to 20} ξ = {numbers between 1 and 20 inclusive}

492  Maths Quest 9

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STATistics and probability

REASONING

One hundred Year 9 Maths students were asked to indicate their favourite topic in mathematics. Sixty chose Probability, 50 chose Measurement and 43 chose Algebra. Some students chose two topics: 15 chose Probability and Algebra, 18 chose Measurement and Algebra, and 25 chose Probability and Measurement. Five students chose all three topics. ξ a Copy and complete the Venn diagram at right. b How many students chose Probability only? Probability 20 Measurement c How many students chose Algebra only? d How many students chose Measurement only? 5 e How many students chose any two of the three topics? A student is selected at random from this group. Find Algebra the probability that this student has chosen: f Probability g Algebra h Algebra and Measurement i Algebra and Measurement but not Probability j all of the topics. 14 Create a Venn diagram using two circles to accurately describe the relationships between the following quadrilaterals: rectangle, square and rhombus. 15 Use the Venn diagram at right to write the numbers 8 ξ A B of the correct regions for each of the following 4 problems. 1 5 a A′ ∪ (B′ ∩ C) b A ∩ (B ∩ C′) 3 c A′ ∩ (B′ ∪ C′) d (A ∪ B ∪ C)′ 2 6 16 A recent survey taken at a cinema asked 90 teenagers what they thought about three different movies. In 7 C total, 47 liked ‘Hairy Potter’, 25 liked ‘Stuporman’ and 52 liked ‘There’s Something About Fred’. 16 liked ‘Hairy Potter’ only. 4 liked ‘Stuporman’ only. 27 liked ‘There’s Something About Fred’ only. There were 11 who liked all three films and 10 who liked none of them. a Construct a Venn diagram showing the results of the survey. b What is the probability that a teenager chosen at random liked ‘Hairy Potter’ and ‘Stuporman’ but not ‘There’s Something About Fred’?

SA

M PL

E

EV

AL U

AT IO

N

O

N LY

13

PROBLEM SOLVING 17

120 children attended a school holiday program during September. They were asked to select their favourite board game from Cluedo, Monopoly and Scrabble. They all selected at least one game, and 4 children chose all three games. In total, 70 chose Monopoly and 55 chose Scrabble.

Topic 13 • Probability  493

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STATistics and probability

EV

AL U

AT IO

N

O

N LY

Some children selected exactly two games — 12 chose Cluedo and Scrabble, 15 chose Monopoly and Scrabble, and 20 chose Cluedo and Monopoly. a Draw a Venn diagram to represent the children’s selections. b What is the probability that a child selected at random did not choose Cluedo as a favourite game? 18 Valleyview High School offers three sports at Year 9: baseball, volleyball and soccer. There are 65 students in Year 9. 2 have been given permission not to play sport due to injuries and medical conditions. 30 students play soccer. 9 students play both soccer and volleyball but not baseball. 9 students play both baseball and soccer (including those who do and don’t play volleyball). 4 students play all three sports. 12 students play both baseball and volleyball (including those who do and don’t play soccer). The total number of players who play baseball is 1 more than the total of students who play volleyball. a Determine the number of students who play volleyball. b If a student was selected at random, what is the probability that this student plays soccer and baseball only? 19 A Venn diagram consists of overlapping ovals which are used to show the relationships between sets. Consider the numbers 156 and 520. Show how a Venn diagram could be used to determine their: a HCF b LCM. 20 A group of 200 shoppers was asked which type of fruit they had bought in the last week. The results are shown in the table.

SA

M PL

E

Fruit

Apples (A) only Bananas (B) only Cherries (C) only A and B A and C B and C A and B and C

Number of shoppers

45 34 12 32 15 26 11

Display this information in a Venn diagram. Calculate n(A ∩ B′ ∩ C). c How many shoppers purchased apples and bananas but not cherries? d Calculate the relative frequency of shoppers who purchased: i apples ii bananas or cherries. e Estimate the probability that a shopper purchased cherries only. a

b

494  Maths Quest 9

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STATiSTicS And probAbiliTy

cHAllEnGE 13.1

N LY

13.5 Two-step experiments The sample space

int-2772

Bag 1

AL U

AT IO

N

O

• Imagine two bags (that are not transparent) that contain coloured counters. The first bag has a mixture of black and white counters, and the second bag holds red, green and yellow counters. In a probability experiment, one counter is to be selected at random from each bag and its colour noted. • The sample space for this experiment can be found using a table called an array that systematically displays all the outcomes.

Bag 2

Bag 2 G

Y

B

BR

BG

BY

W

WR

WG

WY

EV

R

E

Bag 1

SA

M PL

The sample space, ξ = {BR, BG, BY, WR, WG, WY}. • The sample space can also be found using a tree diagram. First selection

B

W

Second selection

Sample space

R

BR

G

BG

Y

BY

R

WR

G

WG

Y

WY

Topic 13 • Probability 495

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STATiSTicS And probAbiliTy

WorKEd EXAmplE 11

WriTE/drAW

Draw an array (a table) showing all the possible outcomes.

2 1, 2 2, 2 3, 2 4, 2 5, 2 6, 2

O

1 2 3 4 5 6

1 1, 1 2, 1 3, 1 4, 1 5, 1 6, 1

AT IO

First die

Second die

3 1, 3 2, 3 3, 3 4, 3 5, 3 6, 3

N

THinK

N LY

Two dice are rolled and the numbers uppermost are noted. List the sample space in an array. a How many outcomes are there? b How many outcomes contain at least one 5? c What is P(at least one 5)?

4 1, 4 2, 4 3, 4 4, 4 5, 4 6, 4

a

The table shows 36 outcomes.

b

Count the outcomes that contain 5. The cells are shaded in the table.

b

Eleven outcomes include 5.

c

There are 11 favourable outcomes and 36 in total.

c

P(at least one 5) = 11 36

There are 36 outcomes.

EV

E

M PL

WorKEd EXAmplE 12

6 1, 6 2, 6 3, 6 4, 6 5, 6 6, 6

AL U

a

5 1, 5 2, 5 3, 5 4, 5 5, 5 6, 5

SA

Two coins are tossed and the outcomes are noted. Show the sample space on a tree diagram. a How many outcomes are there? b Find the probability of tossing at least one Head.

THinK 1

Draw a tree representing the outcomes for the toss of the first coin

WriTE/drAW First coin H

T

496

Maths Quest 9

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STATiSTicS And probAbiliTy

2

For the second coin the tree looks like this:

First coin

Second coin H

Second coin

Sample space

H

HH

T

HT

H

TH

H

T

T

List the outcomes.

TT

Count the outcomes in the sample space.

a

There are 4 outcomes (HH, HT, TH, TT).

b

Three outcomes have at least one Head.

b

P(at least one Head) = 34

N

O

a

Two-step experiments • When a coin is tossed, P(H) = 12, and when a die is rolled, P(3) = 16.

AL U

If a coin is tossed and a die is rolled, what is the probability of getting a Head and a 3? • Consider the sample space.

AT IO

3

N LY

T

Add this tree to both ends of the first tree.

2

3

4

5

6

H

H, 1

H, 2

H, 3

H, 4

H, 5

H, 6

T

T, 1

T, 2

T, 3

T, 4

T, 5

T, 6

EV

1

1 . There are 12 outcomes, and P(Head and 3) = 12

E

1 = 12 × 16. • In this case, P(Head and 3) = P(H) × P(3); that is, 12

SA

M PL

• In general, if A is the outcome of one event and B is the outcome of a separate event, then P(A ∩ B) = P(A) × P(B).

WorKEd EXAmplE 13

In one cupboard Joe has 2 black t-shirts and 1 yellow one. In his drawer there are 3 pairs of white socks and 1 black pair. If he selects his clothes at random, what is the probability that his socks and t-shirt will be the same colour? THinK

WriTE

If they are the same colour then they must be black.

P(Bt ∩ Bs) = P(Bt) × P(Bs)

P(black t-shirt) = P(Bt) = 23

= 23 ×

P(black socks) = P(Bs) = 14

= 16

1 4

Topic 13 • Probability 497

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STATISTICS AND PROBABILITY

Choosing with replacement • Consider what happens when replacement is allowed in an experiment. Worked example 14 illustrates this situation. WORKED EXAMPLE 14

1

Draw a tree for the first trial. Write the probability on the branch. Note: The probabilities should sum to 1.

R

3 5

O

a

WRITE/DRAW

N

THINK

AT IO

2 5

For the second trial the tree is the same. Add this tree to both ends of the first tree.

3 5

2 5

For both draws P(R) = 35 and P(B) = 25. Use the rule P(A ∩ B) = P(A) × P(B) to determine the probabilities.

SA

M PL

E

b

List the outcomes.

EV

3

b

B

3 5

R

AL U

2

N LY

A bag contains 3 red and 2 blue counters. A counter is taken at random from the bag, its colour is noted, then it is returned to the bag and a second counter is chosen. a Show the outcomes on a tree diagram. b Find the probability of each outcome. c Find the sum of the probabilities.

B

R 2 3 5 B 5 R

RR

B

BB

2 5

RB BR

P(R ∩ R) = P(R) × P(R) = 35 × 35 9 = 25

P(R ∩ B) = P(R) × P(B) = 35 × 25 6 = 25

P(B ∩ R) = P(B) × P(R) = 25 × 35 6 = 25

P(B ∩ B) = P(B) × P(B) = 25 × 25 4 = 25

c

498

Add the probabilities.

9

c 25

6 6 4 + 25 + 25 + 25 =1

Maths Quest 9

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STATISTICS AND PROBABILITY

• In Worked example 14, P(R) = 35 and P(B) = 25 for both trials. This would not be so if a counter is selected but not replaced.

Choosing without replacement

If the first counter randomly selected is red, then the sample space for the second draw looks like this:

N LY

• Let us consider again the situation described in Worked example 14, and consider what happens if the first marble is not replaced. • Initially the bag contains 3 red and 2 blue counters, and either a red counter or a blue counter will be chosen. P(R) = 35 and P(B) = 25. • If the counter is not replaced, then the sample space is affected as follows:

WORKED EXAMPLE 15

So P(R) = 34 and P(B) = 14.

EV

So P(R) = 24 and P(B) = 24.

AL U

AT IO

N

O

If the first counter randomly selected is blue, then the sample space for the second draw looks like this:

SA

THINK

M PL

E

A bag contains 3 red and 2 blue counters. A counter is taken at random from the bag and its colour is noted, then a second counter is drawn, without replacing the first one. a Show the outcomes on a tree diagram. b Find the probability of each outcome. c Find the sum of the probabilities.

a

Draw a tree diagram, listing the probabilities.

WRITE/DRAW 2 4

a 3 5

RR

B

RB

R

BR

B

BB

R 2 4

2 5

R

3 4

B 1 4

Topic 13 • Probability 499

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STATiSTicS And probAbiliTy

b

Use the rule P(A ∩ B) = P(A) × P(B) to determine the probabilities.

b

P(R ∩ R) = P(R) × P(R) = 35 ×

2 4

6 = 20 3 = 10

P(R ∩ B) = P(R) × P(B) 6 = 20 3 = 10

2 4

N LY

= 35 ×

P(B ∩ R) = P(B) × P(R) 3 4

O

= 25 ×

AT IO

N

6 = 20

3 = 10

Add the probabilities.

3

c 10

= 25 ×

1 4

2 = 20 1 = 10

3 3 1 + 10 + 10 + 10 =1

EV

c

AL U

P(B ∩ B) = P(B) × P(B)

E

Exercise 13.5 Two-step experiments individUAl pATHWAyS

500



prAcTiSE

M PL

SA

rEFlEcTion How does replacement affect the probability of an event occurring?

Questions: 1–10, 12



conSolidATE



Questions: 1–15 ⬛ ⬛ ⬛ Individual pathway interactivity

mASTEr

Questions: 1–17 int-4537

FlUEncy 1

In her cupboard Rosa has 3 scarves (red, blue and pink) and 2 beanies (brown and purple). If she randomly chooses 1 scarf and 1 beanie, show the sample space in an array.

Maths Quest 9

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STATistics and probability

2

If two dice are rolled and their sum is noted, complete the array below to show the sample space.   WE11 

Die 1 1

3

4

6

2

2

7

3 4

O

Die 2

5

N LY

1

2

5 6

AT IO

N

9

What is P(rolling a total of 5)? b What is P(rolling a total of 1)? c What is the most probable outcome? 3 One box contains red and blue pencils, and a second box contains red, blue and green pencils. If one pencil is chosen at random from each box and the colours are noted, draw a tree diagram to show the sample space.

A bag contains 3 discs labelled 1, 3 and 5, and another bag contains two discs, labelled 2 and 4, as shown below. A disc is taken from each bag and the larger number is recorded.

SA

4

M PL

E

EV

AL U

a

  WE12 

5 1

3

2

4

Topic 13 • Probability  501

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STATistics and probability

a

Complete the tree diagram below to list the sample space. 2

2

4

4

2

3

1

N LY

3

What is: i P(2) ii P(1) iii P(odd number)? 5 Two dice are rolled and the difference between the two numbers is found. a Use an array to find all the outcomes. b Find: i P(odd number) ii P(0) iii P(a number more than 2) iv P(a number no more than 2). 6   WE13  A die is rolled twice. What is the probability of rolling: a a 6 on the first roll  b a double 6 c an even number on both dice  d a total of 12? 7 A coin is tossed twice. a Show the outcomes on a tree diagram. b What is: i P(2 Tails) ii P(at least 1 Tail)? 8   WE14  A bag contains 3 red counters and 1 blue counter. A counter is chosen at random. A second counter is drawn with replacement. a Show the outcomes and probabilities on a tree diagram. b Find the probability of choosing: i a red counter then a blue counter ii two blue counters. 9   WE15  A bag contains 3 black balls and 2 red balls. If two balls are selected, randomly, without replacement: a show the outcomes and their probabilities on a tree diagram b find P(2 red balls).

SA

M PL

E

EV

AL U

AT IO

N

O

b

Understanding

The kings and queens from a deck of cards are shuffled, then 2 cards are chosen. Find the probability that 2 kings are chosen: a if the first card is replaced b if the first card is not replaced. 11 Each week John and Paul play 2 sets of tennis against each other. They each have an equal chance of winning the first set. When John wins the first set, his 10

502  Maths Quest 9

c13Probability.indd 502

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STATiSTicS And probAbiliTy

probability of winning the second set rises to 0.6, but if he loses the first set, he has only a 0.3 chance of winning the second set. a Show the possible outcomes on a tree diagram. b What is: i P(John wins both sets) ii P(Paul wins both sets) iii P(they win 1 set each)?

N LY

rEASoninG

A bag contains 4 red and 6 yellow balls. If the first ball drawn is yellow, explain the difference in the probability of drawing the second ball if the first ball was replaced compared to not being replaced. 13 Three dice are tossed and the total is recorded. a What are the smallest and largest possible totals? b Calculate the probabilities for all possible totals.

N

O

12

AT IO

problEm SolvinG

You draw two cards, one after the other without replacement, from a deck of 52 cards. a What is the probability of drawing two aces? b What is the probability of drawing two face cards (J, Q, K)? c What is the probability of getting a ‘pair’? (22, 33, 44 … QQ, KK, AA)? 15 A chance experiment involves flipping a coin and rolling two dice. Determine the probability of obtaining Tails and two numbers whose sum is greater than 4. 16 In a jar there are 10 red balls and 6 green balls. Jacob takes out two balls, one at a time, without replacing them. What is the probability that both balls are the same colour? 17 In the game of ‘Texas Hold’Em’ poker, 5 cards are progressively placed face up in the centre of the table for all players to use. At one point in the game there are 3 face-up cards (two hearts and one diamond). You have 2 diamonds in your hand for a total of 3 diamonds. Five diamonds make a flush. Given that there are 47 cards left, what is the probability that the next two face-up cards are both diamonds?

M PL

E

EV

AL U

14

SA

cHAllEnGE 13.2

doc-6314

Topic 13 • Probability 503

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STATiSTicS And probAbiliTy

13.6 Mutually exclusive and independent events Mutually exclusive events

AT IO

N

O

N LY

• If two events cannot both occur at the same time then it is said the two events are mutually exclusive. For example, when rolling a die, the events ‘getting a 1’ and ‘getting a 5’ are mutually exclusive. • If two sets are disjoint (have no elements in common), then the sets are mutually exclusive. For example, if A = {prime numbers > 10} and B = {even numbers}, then A and B are mutually exclusive. • If A and B are two mutually exclusive events (or sets), ξ then P(A ∩ B) = ø. A B • Consider the Venn diagram shown. Since A and 2 4 B are disjoint, then A and B are mutually 1 6 5 exclusive sets. 3 • If two events A and B are mutually exclusive, then P(A or B) = P(A ∪ B) = P(A) + P(B).

Examples of mutually exclusive events

AL U

• Draw a card from a standard deck: the drawn card is a heart or a club. – Reason: it is impossible to get both a heart and a club at the same time. • Record the time of arrival of overseas flights: a flight is late, on time or it is early. – Reason: it is impossible for the flight to arrive late, on time or early all at the same time.

EV

Examples of non-mutually exclusive events

M PL

E

• Draw a card from a standard deck: the drawn card is a heart or a king. – Reason: it is possible to draw the king of hearts. • Record the mode of transport of school students: count students walking or going by bus. – Reason: a student can walk (to the bus stop) and take a bus.

WorKEd EXAmplE 16

SA

A card is drawn from a pack of 52 cards. What is the probability that the card is a diamond or a spade? THinK

WriTE

1

The events are mutually exclusive because diamonds and spades cannot be drawn at the same time.

The two events are mutually exclusive as P(A ∩ B) = ∅.

2

Determine the probability of drawing a diamond and the probability of drawing a spade.

Number of diamonds, n(E1) = 13 Number of spades, n(E2) = 13 Number of cards, n(S) = 52 P(spade) = 13 P(diamond) = 13 52 52 = 14

504

= 14

Maths Quest 9

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STATiSTicS And probAbiliTy

3

Write the probability.

4

Evaluate and simplify.

P(A ∪ B) = P(A) + P(B) P(diamond or spade) = P(diamond) + P(spade) = 14 + 14 = 12

N LY

Independent events

O

• Two events are considered independent if the outcome of one event is not dependent on the outcome of the other event. • For example, if E1 = {rolling a 4 on a first die} and E2 = {rolling a 2 on a second die}, the outcome of event E1 is not influenced by the outcome of event E2, so the events are independent.

N

WorKEd EXAmplE 17

THinK

Link each outcome of the first flip with the outcomes of the second part of the experiment (flipping the second coin). Link each outcome from the second flip with the outcomes of the third part of the experiment (flipping the third coin).

SA

M PL

E

3

Use branches to show the individual outcomes for the first part of the experiment (flipping the first coin).

AL U

2

WriTE/drAW

EV

1

AT IO

Three coins are flipped simultaneously. Draw a tree diagram for the experiment. Calculate the following probabilities. a P(3 Heads) b P(2 Heads) c P(at least 1 Head) First coin 1– 2

1– 2

H

T

First coin

1– 2

1– 2

Second coin

H

T

First coin

1– 2

H

1– 2

T

1– 2

H

1– 2

T

Second coin

Third coin 1– 2

1– 2

1– 2

H

T

1– 2

H

1– 2

T

1– 2 1– 2

1– 2 1– 2

H

1– 2

T 1– 2

H T

1– 2

H T

1– 2

H T

1– 2

H T

Topic 13 • Probability 505

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STATistics and probability

Determine the probability of each outcome. Note: The probability of each result is found by multiplying along the branches and in each case this will be 12 × 12 × 12 = 18.

1

1– 2

1– 2

H

T

2 1– 2

H

1– 2

T

1– 2 1– 2

1– 2 1– 2 1– 2

H

1– 2

T

1– 2

1– 2

1– 2

1– 2

2

Write your answer.

1

At least 1 Head means any outcome that contains one or more Head. This is every outcome except three Tails. That is, it is the complementary event to obtaining 3 Tails.

1– 2

×

1– 2

=

H

HTH

×

H

THH

T

THT

H

TTH

1– 2 1– 2 1– 2 1– 2 1– 2 1– 2

=

HTT

1– 2 1– 2 1– 2 1– 2 1– 2 1– 2

×

T

1– 2 1– 2 1– 2 1– 2 1– 2 1– 2

T

× × × ×

× × × ×

TTT

P(2 Heads)   = P(H, H, T) + P(H, T, H) + P(T, H, H)   = 18 + 18 + 18

×

= = = =

— 1

O

b

×

=

1– 8 1– 8 1– 8 1– 8 1– 8 1– 8

The probability of obtaining exactly 2 Heads is 38.

E

Write your answer.

×

P(3 Heads) = 18.

  = 38

M PL

2

1– 2

N

{2 Heads} has 3 satisfactory outcomes: (H, H, T), (H, T, H) and (T, H, H), which are mutually exclusive.

HHT

a

c

EV

c

1

T

1– 8 1– 8

AT IO

b

The probability of three heads is P(H, H, H)

Outcomes Probability 1– 1– 1– HHH 2 × 2 × 2 =

AL U

a

3 H

N LY

4

P(at least 1 Head)   = 1 − P(T, T, T)   = 1 − 18   = 78

The probability of obtaining at least 1 Head is 78.

SA

Note: The probabilities of all outcomes add to 1.

Dependent events •• Many real-life events have some dependence upon each other, and their probabilities are likewise affected. Examples include: –– the chance of rain today and the chance of a person taking an umbrella to work –– the chance of growing healthy vegetables and the availability of good soil –– the chance of Victory Soccer Club winning this week and winning next week –– drawing a card at random, not replacing it, and drawing another card. •• It is important to be able to recognise the difference between dependent events and independent events.

506  Maths Quest 9

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STATiSTicS And probAbiliTy

WorKEd EXAmplE 18

A jar contains three black marbles, five red marbles, and two white marbles. Find the probability of choosing a black marble (with replacement), then choosing another black marble. THinK

The events, draw 1 and draw 2, are independent E1 and E2 are independent events. because the result of the first draw is not dependent 3 –– B 10 on the result of the second draw. 5 –– 10

B

2

3 –– 10

Demonstrate using a tree diagram.

2 –– 10 3 –– 10

5 –– 10

W

5 –– 10

B R

O

R

R

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1

WriTE/drAW

3

AT IO

2 –– 10

3 –– 10

W

W

5 –– 10

2 –– 10

B R W

P(E1 and E2) = P(E1) × P(E2) P(black and black) = P(black) × P(black)

Determine the probability.

3 P(black and black) = 10 ×

3 10

9 = 100

EV

AL U

Evaluate and simplify.

N

2 –– 10

E

• If the first marble had not been replaced in the previous worked example, the second draw would be dependent on the outcome of the first draw, and so it follows that the sample space for the second draw is different from that for the first draw.

M PL

WorKEd EXAmplE 19

Repeat Worked example 15 without replacing the first marble before the second one is drawn. THinK

The words ‘without replacing’ indicate that the two events are dependent. Write the sample space and state the probability of choosing a black marble on the first selection.

There are 10 marbles and 3 of these are black. The sample space is {B, B, B, R, R, R, R, R, W, W}.

Assume that a black marble was chosen in the first selection. Determine how many black ones remain, and the total number of remaining marbles. Write the sample space and state the probability of choosing a black marble on the second selection.

A black one was chosen, leaving 2 black ones and a total of 9 marbles. The sample space is {B, B, R, R, R, R, R, W, W}.

SA

1

2

WriTE/drAW

3 P(E1) = P(black) = 10

P(E2) = P(black) = 29

Topic 13 • Probability 507

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STATiSTicS And probAbiliTy

Demonstrate using a tree diagram.

2– 9

B 3 –– 10

2– 9 3– 9

5 –– 10

R

5– 9

3– 9

W

4– 9

B R W

5– 9

1– 9

B R

O

W

P(E1 and E2) = P(E1) × P(E2) 3 × 29 = 10

Multiply the probabilities.

N

4

R W

2– 9 2 –– 10

B

N LY

3

5

AT IO

1 = 15

Answer the question.

The probability of choosing two black marbles 1 . without replacing the first marble is 15

AL U

Exercise 13.6 Mutually exclusive and independent events ⬛

rEFlEcTion What is the difference between independent events and mutually exclusive events?

prAcTiSE

EV

individUAl pATHWAyS

M PL

E

Questions: 1–3, 5, 7, 9, 11, 13–16, 18, 20, 22, 24, 26, 29



conSolidATE



Questions: 1, 3–6, 8, 10, 12–15, 18, 20, 24–26, 28, 29

⬛ ⬛ ⬛ Individual pathway interactivity

mASTEr

Questions: 1, 5–8, 11, 13, 14, 16, 17, 19–23, 25, 27, 28, 30

int-4538

FlUEncy

SA

1

508

If a card is drawn from a pack of 52 cards, what is the probability that the card is not a queen? MC

A

4 52

b

4 48

c

13 12

d

48 52

Which events are not mutually exclusive? A Drawing a queen and drawing a jack from 52 playing cards b Drawing a red card and drawing a black card from 52 playing cards c Drawing a vowel and drawing a consonant from cards representing the 26 letters of the alphabet d Obtaining a total of 8 and rolling doubles (when rolling two dice) 1 3 When a six-sided die is rolled 3 times, the probability of getting 3 sixes is . What is 216 the probability of not getting 3 sixes? 2

MC

Maths Quest 9

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STATistics and probability

Eight athletes compete in a 100-m race. The probability that the athlete in lane 1 will win is 15. What is the probability that one of the other athletes wins? (Assume that there are no dead heats.)

4   MC 

1 5 B 5 8 8 4 C D 5 5

N LY

A

pencil case has 4 red pens, 3 blue pens and 5 black pens. If a pen is randomly drawn from the pencil case, find: a P(drawing a blue pen) b P(not drawing a blue pen). 6 Seventy Year 9 students were surveyed. Their ages ranged from 13 years to 15 years, as shown in the table below. 13 10  7 17

14 20 15 35

15  9  9 18

AT IO

Age Boys Girls Total

N

O

5 A

Total 39 31 70

SA

M PL

E

EV

AL U

A student from the group is selected at random. Find: a P(selecting a student of the age of 13 years) b P(not selecting a student of the age of 13 years) c P(selecting a 15-year-old boy) d P(not selecting a 15-year-old boy). 7   WE16  A card is drawn from a pack of 52 cards. What is the probability that the card is a king or an ace? 8   MC  A die is rolled. Find the probability of getting an even number or a 3. 3 4 1 5 4 1 A B C D 6 6 6 6 9 If you spin the following spinner, what is the probability of obtaining: 3 2 a a 1 or a 3 b an even number or an odd number? 10 The probabilities of Dale placing 1st, 2nd, 3rd or 4th in the local surf competition are: 7 . 1st = 16 2nd = 15 3rd = 25 4th = 30 Find the probability that Dale places: a 1st or 2nd b 3rd or 4th c 1st, 2nd or 3rd d not 1st. 11   WE17  A circular spinner that is divided into two equal halves, coloured red and blue, is spun 3 times. a Draw a tree diagram for the experiment. b Calculate the following probabilities.  i P(3 red sectors) ii  P(2 red sectors) iii P(1 red sector) iv P(0 red sectors) v  P(at least 1 red sector) vi P(at least 2 red sectors) Topic 13 • Probability  509

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STATistics and probability

13 14 15 16

There are two yellow tickets, three green tickets, and four black tickets in a jar. Choose one ticket, replace it, then choose another ticket. Find the probability that a yellow ticket is drawn first, then a black ticket.   WE19  Repeat question 12 with the first ticket not being replaced before the second ticket is drawn. A coin is tossed two times. Determine P(a Head and a Tail in any order). A coin is tossed three times. Determine P(H, H, T) (in that order). A coin and a die are tossed. What is the probability of a Heads–2 outcome?   WE18 

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12

UNDERSTANDING

M PL

E

21

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20

EV

19

AT IO

N

18

Holty is tossing two coins. He claims that flipping two Heads and flipping zero Heads are complementary events. Is he correct? Explain your answer. Each of the numbers 1, 2, 3, .  .  . 20 is written on a card and placed in a bag. If a card is drawn from the bag, find: a P(drawing a multiple of 3 or a multiple of 10) b P(drawing an odd number or a multiple of 4) c P(drawing a card with a 5 or a 7) d P(drawing a card with a number less than 5 or more than 16). From a shuffled pack of 52 cards, a card is drawn. Find: a P(hearts or the jack of spades) b P(a queen or a jack) c P(a 7, a queen or an ace) d P(neither a club nor the king of spades).   MC  Which are not mutually exclusive? A Obtaining an odd number on a die and obtaining a 4 on a die B Obtaining a Head on a coin and obtaining a Tail on a coin C Obtaining a red card and obtaining a black card from a pack of 52 playing cards D Obtaining a diamond and obtaining a king from a pack of 52 playing cards Greg has a 30% chance of scoring an A on an exam, Carly has 70% chance of scoring an A on the exam, and Chilee has a 90% chance of scoring an A on the exam. What is the probability that all three can score an A on the exam? From a deck of playing cards, a card is drawn at random, noted, replaced and another card is drawn at random. Find the probability that: a both cards are spades b neither card is a spade c both cards are aces d both cards are the ace of spades e neither card is the ace of spades. Repeat question 22 with the first drawn card not being replaced before the second card is drawn. Assuming that it is equally likely that a boy or a girl will be born, answer the following. a Show the gender possibilities of a 3-child family on a tree diagram. b In how many ways is it possible to have exactly 2 boys in the family? c What is the probability of getting exactly 2 boys in the family? d Which is more likely, 3 boys or 3 girls in the family? e What is the probability of having at least 1 girl in the family?

O

17

SA

22

23 24

510  Maths Quest 9

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STATistics and probability

REASONING

Give an example of mutually exclusive events that are not complementary events using: a sets b a Venn diagram. 26 Explain why all complementary events are mutually exclusive but not all mutually exclusive events are complementary. 27 A married couple plans to have four children. a List the possible outcomes in terms of boys and girls. b What is the probability of them having exactly two boys? c Another couple plans to have two children. What is the probability that they have exactly one boy?

N LY

25

PROBLEM SOLVING

A bag contains 6 marbles, 2 of which are red, 1 is green and 3 are blue. A marble is drawn, the colour is noted, the marble is replaced and another marble is drawn. a Show the possible outcomes on a tree diagram. b List the outcomes of the event ‘the first marble is red’. c Calculate P(the first marble is red). d Calculate P(2 marbles of the same colour are drawn). 29 A tetrahedron (prism with 4 identical triangular faces) is numbered 1, 1, 2, 3 on its 4 faces. It is rolled twice. The outcome is the number facing downwards. a Show the results on a tree diagram. b Are the outcomes 1, 2 and 3 equally likely? c Find the following probabilities: i P(1, 1)   ii P(1 is first number) iii P(both numbers the same) iv P(both numbers are odd). 30 Robyn is planning to watch 3 footy games on one weekend. She has a choice of two games on Friday night: (A) Carlton vs West Coast and (B) Collingwood vs Adelaide. On Saturday, she can watch one of three games: (C) Geelong vs Brisbane, (D) Melbourne vs Fremantle and (E) North Melbourne vs Western Bulldogs. On Sunday, she also has a choice of three games: (F) St Kilda vs Sydney, (G) Essendon vs Port Adelaide and (H) Richmond vs Hawthorn. She plans to watch one game each day and will choose a game at random. a To determine the different combinations of games Robyn can watch, she draws a tree diagram using codes A, B, .  .  . H. List the sample space for Robyn’s selections. b Robyn’s favourite team is Carlton. What is the probability that one of the games Robyn watches involves Carlton? c Robyn has a good friend who plays for St Kilda. What is the probability that Robyn watches both the matches involving Carlton and St Kilda?

SA

M PL

E

EV

AL U

AT IO

N

O

28

Topic 13 • Probability  511

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STATiSTicS And probAbiliTy

13.7 Conditional probability • The probability that an event occurs given that another event has already occurred is called conditional probability. • The probability that event B occurs, given that event A has already occurred is denoted by P(B | A). The symbol ‘|’ stands for ‘given’. • The formula for conditional probability is: P(A ∩ B) , P(A) ≠ 0. P(A)

N LY

P(B | A) = WorKEd EXAmplE 20

12

ξ

B 7

10

N

A

O

This Venn diagram below shows the results of a survey where students were asked to indicate whether they liked apples or bananas.

AT IO

4

AL U

If one student is selected at random: a What is the probability that the student likes bananas? b What is the probability that the student likes bananas, given that they also like apples? c Comment on any differences between the answers for parts a and b. THinK

Find the total number of students.

2

Find the total number of students who like bananas.

3

Determine the probability using the correct formula.

a

EV

1

SA

b

E

4

Write the answer.

1

Determine the number of students who like apples.

2

Find the probability that a student likes apples.

Total number of students in survey = 12 + 7 + 10 + 4 = 33 Total number who like bananas = 7 + 10 = 17 P(bananas) = P(B) Total number who like bananas = Total number of students

M PL

a

WriTE

= 17 33 The probability that a student likes . bananas is 17 33 b

Number of students who like apples = 12 + 7 = 19 P(apples) = P(A) =

Number of students who like apples Total number of students

= 19 33 512

Maths Quest 9

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STATiSTicS And probAbiliTy

3

Note the number liking both apples and bananas. This is the overlapping region of the two sets.

Number who like both apples and bananas = n(A ∩ B) =7

4

Determine the probability a student likes both apples and bananas.

7 P(A ∩ B) = 33

5

Apply the formula to determine the conditional probability.

P(B | A) =

=

The probability that a student likes bananas, given that they also like apples 7 is 19 . This answer is also supported by the figures in the Venn diagram.

O

Write the answer.

Why aren’t the answers for parts a and b both the same?

c

The answer for part a determines the proportion of students who like bananas out of the whole group of students. The part b answer gives the proportion of students who like bananas out of those who like apples.

AL U

c

AT IO

N

6

7 33 19 33 7 19

N LY

=

P(A ∩ B) P(A)

EV

Note: These probabilities could also be expressed as decimals or percentages. • It is possible to transpose the conditional probability formula to determine P(A ∩ B). P(A ∩ B) = P(A) × P(B | A)

E

WorKEd EXAmplE 21

SA

THinK

M PL

In a student survey, the probability that a student likes apples is 19 . The 33 7 . probability that a student likes bananas, given that they also like apples, is 19 What is the probability that a student selected at random likes both apples and bananas? 1

2

3

Write the given information.

WriTE

P(A) = 19 33 7 P(B | A) = 19

Apply the rearranged conditional probability formula.

P(A ∩ B) = P(A) × P(B | A)

Answer the question.

The probability that a student selected at  7 . random likes both apples and bananas is 33

× = 19 33

7 19

7 = 33

Topic 13 • Probability 513

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STATiSTicS And probAbiliTy

• Conditional probability can also be determined by examining outcomes from a tree diagram.

WorKEd EXAmplE 22

Draw a tree diagram to display the flipping of three coins. Write the individual outcomes.

a

1

2 H

H

T

AT IO

N

a

WriTE/drAW

O

THinK

N LY

Three coins are flipped simultaneously. a Display the outcomes as a tree diagram. b Determine the probability that a Head will result from the third coin, given that the first two coins resulted in a Head (H) and a Tail (T).

T

SA

T

Outcomes

H

HHH

T

HHT

H

HTH

T

HTT

H

THH

T

THT

H

TTH

T

TTT

1

Look for the outcomes where the first two flips resulted in a Head and a Tail.

2

How many of these outcomes have a Head for the third flip?

There are two of these outcomes where the third flip resulted in a Head — HTH and THH.

3

Calculate the probability.

From four possible outcomes, two satisfy the conditions. P(H on third flip | H and T on first two flips) = 24

b

There are four outcomes where the first two flips are a Head and a Tail — HTH, HTT, THH and THT.

AL U

EV

E

M PL

b

H

3

= 12

• A two-way table can also be used to determine conditional probability.

WorKEd EXAmplE 23

Two dice are rolled and the numbers are added. a Show the results in a two-way table. b Determine the probability that the sum of the two dice is 7, given that their total is greater than 6.

514

Maths Quest 9

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STATiSTicS And probAbiliTy

THinK

WriTE/diSplAy

Show the results of rolling two dice in a two-way table.

a

a

Die 2 2

1 1

3

4

5

6

(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)

2 (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)

4 (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) 5 (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)

1

Which outcomes have a total greater than 6?

2

Which of these outcomes have a total equal to 7?

There are 6 of these outcomes that have a total of 7 − (6, 1), (5, 2), (4, 3), (3, 4), (2, 5), (1, 6).

3

Write the probability.

6 = 27 P(total of 7 | total greater than 6) = 21

b

There are 36 outcomes. 21 of these have a total greater than 6 (6, 1), (5, 2), (4, 3), (3, 4), (2, 5), (1, 6), (6, 2) . . . etc.

AL U

AT IO

N

b

O

6 (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)

N LY

Die 1

3 (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)

Exercise 13.7 Conditional probability individUAl pATHWAyS prAcTiSE



conSolidATE

EV



Questions: 1–3, 5, 7, 9–11

M PL

E

int-4539

A group of motocross racers was asked to comment on which of two tracks, A or B, they used. The results were recorded in the Venn diagram below. WE21

SA

1

rEFlEcTion How can you determine when a probability question is a conditional one?

mASTEr

Questions: 1–15

⬛ ⬛ ⬛ Individual pathway interactivity

FlUEncy



Questions: 1–4, 6–13

ξ

A 23

B 16

15 6

How many motocross racers were surveyed? b Calculate P(A ∩ B). c Calculate: i P(A) ii P(B | A). d Calculate: i P(B) ii P(A | B). a

Topic 13 • Probability 515

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STATistics and probability

Consider your answers to question 1. a Use your answers from part c to determine P(A ∩ B). b Use your answers from part d to determine P(A ∩ B). c Comment on your answers to parts a and b in this question. 3 A survey was conducted to determine whether a group of students preferred drink A or drink B. The results of the survey produced the following probabilities. 7 and P(B | A) = 37. P(A) = 10 Determine P(A ∩ B). 4   WE22  Two fair coins are tossed. a Display the outcomes as a tree diagram. b Determine the probability that a Head results on the second coin, given that the first coin also resulted in a Head. 5   WE23  Two standard dice are rolled and the numbers are added together. a Show the results in a two-way table. b Determine the probability that the sum of the two dice is even, given that their total is greater than 7.

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UNDERSTANDING

A group of 40 people was surveyed regarding the types of movies, comedy or drama, that they enjoyed. The results are shown below. 28 enjoyed comedy. 17 enjoyed drama. 11 liked both comedy and drama. 6 did not like either type. a Draw a Venn diagram to display the results of the survey. b Determine the probability that a person selected at random: i likes comedy ii likes drama iii likes both comedy and drama iv likes drama, given that they also like comedy v likes comedy, given that they also like drama. c Arrange the probabilities in part b in order from least probable to most probable. 7 A teacher gave her class two tests. Only 25% of the class passed both tests, but 40% of the class passed the first test. What percentage of those who passed the first test also passed the second test?

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REASONING

If P(A) = 0.3, P(B) = 0.5 and P(A ∪ B) = 0.6, calculate: a P(A ∩ B) b P(B | A) c P(A | B). 9 A group of 80 boys is auditioning for the school musical. They are all able to either sing, play a musical instrument, or both. Of the group, 54 can play a musical instrument and 35 are singers. What is the chance that if a randomly selected student is a singer he can also play a musical instrument?

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516  Maths Quest 9

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STATiSTicS And probAbiliTy

A white die and a black die are rolled. The dice are 6-sided and unbiased. Consider the following events. Event A: the white die shows a 6. Event B: the black die shows a 2. Event C: the sum of the two dice is 4. Determine the following probabilities. a P(A | B) b P(B | A) c P(C | A) d P(C | B)

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A die is rolled and the probability of rolling a 6 is 16. However, with the condition that the number rolled was an even number, its probability is 13. Explain why the probabilities are different, using conditional probability.

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problEm SolvinG

H 35

J 12

29

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A group of students was asked to nominate their favourite form of dance, hip hop (H) or jazz (J ). The results are illustrated in the Venn diagram. Use the Venn diagram given to calculate the following probabilities relating to a student’s favourite form of dance.

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12

14

What is the probability that a randomly selected student likes jazz? b What is the probability that a randomly selected student enjoys hip hop, given that they like jazz? 13 At a school classified as a ‘Music school for excellence’, the probability that a student elects to study Music and Physics is 0.2. The probability that a student takes Music is 0.92. What is the probability, correct to 2 decimal places, that a student takes Physics, given that the student is taking Music? 14 A medical degree requires applicants to participate in two tests, an aptitude test and an emotional maturity test. This year, 52% passed the aptitude test and 30% passed both tests. Use the conditional probability formula to calculate the probability, correct to 2 decimal places, that a student who passed the aptitude test also passed the emotional maturity test. 15 The probability that a student is well and misses a work shift the night before an exam is 0.045, while the probability that a student misses a work shift is 0.05. What is the probability that a student is well, given they miss a work shift the night before an exam?

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doc-6315

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STATiSTicS And probAbiliTy

13.8 Review

www.jacplus.com.au

The Maths Quest Review is available in a customisable format for students to demonstrate their knowledge of this topic.

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Language

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int-2711

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array certain chance chance experiment complementary conditional dependent equally likely event

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int-3212

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Link to assessON for questions to test your readiness For learning, your progress AS you learn and your levels oF achievement. assessON provides sets of questions for every topic in your course, as well as giving instant feedback and worked solutions to help improve your mathematical skills. www.assesson.com.au

Download the Review questions document from the links found in your eBookPLUS.

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A summary of the key points covered and a concept map summary of this topic are available as digital documents.

int-2712

Review questions

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The Review contains: • Fluency questions — allowing students to demonstrate the skills they have developed to efficiently answer questions using the most appropriate methods • problem Solving questions — allowing students to demonstrate their ability to make smart choices, to model and investigate problems, and to communicate solutions effectively.

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ONLINE ONLY

experiment experimental probability favourable outcome impossible independent intersection likely mutually exclusive outcome

probability random sample space scale theoretical probability tree diagram trial two-way table Venn diagram

The story of mathematics is an exclusive Jacaranda video series that explores the history of mathematics and how it helped shape the world we live in today. What are the chances? (eles-1700) takes a look at the history of mathematical probability, then goes on to see how probability plays a crucial role in the modern world to the extent that it saves thousands of lives every year.

Topic 13 • Probability 519

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For ricH TASK or For pUZZlE invESTiGATion

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Sand-rings

B

A

C

520

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STATiSTicS And probAbiliTy

0

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The first sand-rings puzzle requires 8 shells to be arranged inside the circles, so that 4 shells appear inside circle A, 5 shells appear inside circle B and 6 shells appear inside circle C. The overlapping of the circles shows that the shells can be counted in 2 or 3 circles. One possible arrangement is shown below. Use this diagram to answer questions 1 to 4. 1 How many shells ξ appear inside A B circle A, but not circle B? 1 1 0 2 How many shells 1 appear in circles 3 2 B and C, but not circle A?

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A shell is selected at random from the sand. 3 What is the probability it came from circle A? 4 What is the probability it was not in circle C? 5 The class was challenged to find the rest of the arrangements of the 8 shells. (Remember:  4 shells need to appear in circle A, 5 in circle B and 6 in circle C.)

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After completing the first puzzle, the students are given new rules. The number of shells to be arranged in the circles is reduced from 8 to 6. However, the number of shells to be in each circle remains the same; that is, 4 shells in circle A, 5 shells in circle B and 6 shells in circle C. 6 Using 6 shells, in how many ways can the shells be arranged so that there are 4, 5 and 6 shells in the three circles? 7 Explain the system or method you used to determine your answer to question 6 above. Draw diagrams in the space provided to show the different arrangements. 8 Using 7 shells, in how many ways can the shells be arranged so that there are 4, 5 and 6 shells in the three circles? 9 Again, explain the system or method you used to determine your answer to question 8. Draw diagrams in the space provided to show the different arrangements. 10 What would be the minimum number of shells required to play sand-rings, so that there are 4, 5 and 6 shells in the three circles? 11 Modify the rules of this game so that different totals are required for the 3 circles. Challenge your classmates to find all possible solutions to your modified game.

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STATiSTicS And probAbiliTy For ricH TASK or For pUZZlE

codE pUZZlE

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Roy G. Biv is a mnemonic (memory aid) for what purpose? The number of elements in the regions of the Venn diagrams gives the puzzle’s answer code. The numbers shown indicate the number of elements in the region. n (ε ) = 20

P

ε

Y

O

ε

n (P ) = 16

X

Q

20 7

15

8

5

Z

Q )´

=2

a = n (Q ) =

g = n (Y ) =

Q)=



h = n (Z ) =

ε

k = n (ε ) =

n (ε ) = 50

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n (R ) = 38

S

C 8

n (S ) = 21





E



H

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n (ε ) = 45

r = n (B

[A C ]´ ) =

s = n (ε ) =

K

t = n (K ) =

11

v = n (H



K´ ) =

3

z = n (H ) =

7 40 9

522

[A B ]´ ) =



ε 4

q = n (C



S )´ =

n (B ) = 17

B´ ) =

o = n (A C



n = n (R

B

n (A ) = 19

R´ ) =

6

2

4

A

S´ ) =

m = n (S

3

n (C ) = 14



EV

18

l = n (R

ε



d = n (P



P´ ) =



b = n (Q

R

Y´ ) =

e = n (X

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= 5, n (P



Q)



n (P

N

n (X ) = 31

18 9

18 5

11

7

20

18 20 18 20 40

42

18

11 7 5

15 18

5 31

15 2 23

11 9

66 3

8 9 7

15 15 9

4 5

8 27

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STATiSTicS And probAbiliTy

Activities

13.3 Experimental probability digital doc • WorkSHEET 13.1 (doc-6313): Experimental probability interactivity • IP interactivity 13.3 (int-4535): Experimental probability

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13.7 conditional probability digital doc • WorkSHEET 13.3 (doc-6315): Probability III interactivity • IP interactivity 13.7 (int-4539): Conditional probability 13.8 review interactivities • Word search (int-2711) • Crossword (int-2712) • Sudoku (int-3212) digital docs • Topic summary (doc-13667) • Concept map (doc-13668)

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13.4 venn diagrams and two-way tables digital docs • SkillSHEET (doc-6311): Determining complementary events • SkillSHEET (doc-6312): Calculating the probability of a complementary event interactivity • IP interactivity 13.4 (int-4536): Venn diagrams and two-way tables

13.6 mutually exclusive and independent events interactivity • IP interactivity 13.6 (int-4538): Mutually exclusive and independent events

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13.2 Theoretical probability digital docs • SkillSHEET (doc-6307): Probability scale • SkillSHEET (doc-6308): Understanding a deck of playing cards • SkillSHEET (doc-6309): Listing the sample space • SkillSHEET (doc-6310): Theoretical probability interactivities • IP interactivity 13.2 (int-4534): Theoretical probability • Random number generator (int-0089)

13.5 Two-step experiments interactivities • Two-step chance (int-2772) • IP interactivity 13.5 (int-4537): Two-step experiments digital doc • WorkSHEET 13.2 (doc-6314): Probability II

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13.1 overview video • The story of mathematics: What are the chances? (eles-1700)

www.jacplus.com.au

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To access ebookplUS activities, log on to

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STATistics and probability

Answers

topic 13 Probability

0.5

1

2 a d g 3 a c e 4 a

Certain b Certain c Unlikely Likely e Certain f Likely Likely h Likely {1, 2, 3, 4, 5, 6} b {H, T} {defective, not defective} d {red, black} {hearts, clubs, diamonds, spades} {1, 2, 3} b {1, 3, 5} c {1, 2, 4, 6} d {3, 4, 5, 6} e {1, 2, 3, 4} f {5, 6} 5 a 3, 5 b 1, 2 c 6 d 1, 2, 5, 6 e 1, 2, 3, 5 f no favourable outcomes g 3 6 a {hearts, clubs, diamonds, spades} b   i  {clubs, spades} ii  {hearts, diamonds} iii {clubs, diamonds, spades} iv {hearts, clubs, diamonds, spades} 7 a 6 b 2 c 52 d 4 e 5 1 1 1 1 8 a b c d 13 13 52 4

1 1 1 1 1 b c d e 52 2 26 4 2 1 4 2 12 f g h i 0 j 13 52 13 13 2 9 4 7 10 a b c d 11 11 11 11 1 1 5 3 11 a b c d 0 e 4 8 16 4 2 1 12 a i  ii 3 3

EV

b Answers will vary. 13, 14 Check with your teacher. 15 Probabilities must be between 0 and 1, so −0.5 and 1.05 can’t be

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probabilities.

16 The coloured portions outside the arc of the spinner shown are of

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no consequence. The four colours within the arc of the spinner are of equal area (each 14 circle), so there is equal chance of falling on each of the colours. 17

conditions.

e  Not reasonable; monthly rainfall in Sydney is not consistent

throughout the year. 5

59

7 a b 24 72 8 A 9 200 10 20 11 B 12 a

11 3 17 3 b c d 20 10 20 20 33 40 3 b = 25 c 100 100 100

13 a

14 a i  6 ii {356, 365, 536, 563, 635, 653} 2 3

1 3

1 3

b c d 5 D 1 16 Svetlana 6, Sarah 4, Leonie 3, Trang 2 17 C 18 a 27 b {111, 112, 115, 121, 122, 125, 151, 152, 155, 211, 212, 215,

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e b a

2 a i r.f.(H) = 0.44, r.f.(T) = 0.56 ii 1 b i r.f.(H) = 0.62, r.f.(T) = 0.38 ii 1 3 Each answer will be different. 4 a 0.365 b 0.33 c 0.25 d 0.875 5 a 0.275 b 0.64 c 0.365 d 0.635 6 a Not reasonable; not enough trials were held. b Not reasonable; the conditions are different under each trial. c Not reasonable; there are seasonal influences. d  Reasonable; enough trials were performed under the same

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Exercise 13.2  Theoretical probability

2 3

18 There are 36 outcomes, 15 under 7, 6 equal to 7 and 15 over 7.

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So, you would have a greater chance of winning if you chose ‘under 7’ or ‘over 7’ rather than ‘equal to 7’. 9 0.09 1 Exercise 13.3  Experimental probability 1 a   i  16 ii 4 iii 

Outcome

1

2

3

4

Relative 0.19 0.06 0.31 0.13 frequency iv 1 b   i  270 ii 40 iii 

Outcome

1

Relative 0.19 frequency iv 1

5 0.25

6

221, 222, 225, 251, 252, 255, 511, 512, 515, 521, 522, 525, 551, 552, 555} 1 2 3 3 1 1 b 12 20

1 3

c d e 19 a

c No, because John has a higher probability of winning. 20 a

7 10

b, c  Yes, far too many mistakes 21 32 22 1000 balls 23 Red = 10, yellow = 50, blue = 40 7 2 24 a b 20 5 25 a

30 241

b

91 241

c

59 241

Exercise 13.4  Venn diagrams and two-way tables

1 a A′ = {11, 13, 14, 16, 17, 19, 20} b B′ = {20} c C′ = {12, 14, 15, 16, 18, 20} d D′ = {12, 14, 16} 2 a 27 b 11 c 13 d 6 e 6 f 7 g 9 3

T 6 7

S S′

0.06 4 ξ

T′ 5 9

S

2

3

4

5

6

V

0.14

0.17

0.18

0.15

0.17

7

21

2

10

524  Maths Quest 9

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STATistics and probability

5 a i W′ b i A ∩ B ii A ∩ B′ c i A′ ∩ B′ ii A′ ∩ B 6 a

14

Left-handed

Right-handed

Male

15

40

Female

12

33

Rectangle

L

33

16 a

73 2 3 c i  ii iii 100 5 25 8 17 1 3 7 a b c d 25 25 5 25 1 1 31 8 a b c 2 10 40 27 9 d e 40 40 5 3 5 7 9 a 16 b i  ii iii iv 16 16 8 16

3

H

H′

C

3

2

C′

4

3

N 8 90

b 19

1 13

ξ

156

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520 2 2 13

ξ

12

12

b 4 c 21 81 d i  200

13

Algebra 15

e 3 5

b 25 c 15 d 12 e 43 f 43 9 13 1 g h i j 100 50 100 20

34

11 15

C 58

Measurement

5 10

B

21 4

Favourite topic

25

5

A

45

20

2

a HCF = 2 × 2 × 13 = 52 b LCM = 3 × 2 × 2 × 13 × 2 × 5 = 1560 0 a ξ 2

15

Probability

15

7 12

3

C

13 a

4

S 24

E

4 6 8 B 12 2 14 16 10 18 20

M 31

20

18 a 31 students

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1 73 A 9 11 13 17 19 5

C 14

12

3 1 1 7 b i  ii iii iv 12 4 4 2

12

10

4 = 45

ξ

b

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ii 

2

27 Fred

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2

4

11

12

17 a

4

3

S 8

16

b

H C

HP

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0 a Children who are not swimmers 1 b Children who are swimmers and runners c Children who neither swim nor run d Children who swim or run or both i  ξ

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40

11 a

Rhombus

5 a 2, 5, 6, 7, 8 1 b 4 c 5, 7, 8 d 8

12

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Square

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b ξ

Quadrilaterals

ξ

ii

97 200

3 50

Challenge 13.1

120

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STATistics and probability

Bl

Pi

1 4

d

Br

Br, R

Br, Bl

Br, Pi

Pu

Pu, R

Pu, Bl

Pu, Pi

Die 2

1

2

3

4

5

6

1

2

3

4

5

6

7

2

3

4

5

6

7

8

3

4

5

6

7

8

9

4

5

6

7

8

9

10

5

6

7

8

9

10

11

7

8

9

10

11

7 a

b i 

1 2

1 2

H

1 2

T

B

RB

G

RG

R

BR

B

BB

G

BG

8 a

12

3 4

R

1 4

B

2

3

4

4

EV

4

E

5

4

5

1 2

b i  ii 0 iii

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3 5

2 5

b

1 10

10 a

1 4

b

3 14

5

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2

Die 2

TH

1 2

T

TT

3 4

R

1 4 3 4

B

RR RB

R

BR

1 4

B

BB

2 4

B

BB

3 10

2 4 3 4

R

BR

3 10

B

RB

3 10

1 4

R

RR

1 10

0.6

J

JJ

0.4

P

JP

0.3

J

PJ

0.7

P

PP

3 16

9 a

2

3

1 2

H

1 16

4

5 a

HT

ii 

1

1 6

T

3 4

b i 

2

4 a

1 2 1 2

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B

RR

HH

ii 

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R

R

H

1 4

1 a b 0 c A total of 7 9

3

1 36

c

Die 1

6

1 36

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2

b

N

Scarves

Beanies

1 6

6 a

1

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Exercise 13.5  Two-step experiments

11 a 0.5

Die 1

B

R

J

1

2

3

4

5

6

1

0

1

2

3

4

5

2

1

0

1

2

3

4

3

2

1

0

1

2

3

4

3

2

1

0

1

2

b i  0.3 ii  0.35 iii 0.35

5

4

3

2

1

0

1

12 If the first ball is replaced, the probability of drawing a yellow

6

5

4

3

2

1

0

1 6

1 3

2 3

b i  ii iii iv

0.5

P

ball stays the same on the second draw, i.e. (35). If the first ball isn’t replaced, the probability of drawing a yellow ball on the second draw decreases, i.e. (59).

526  Maths Quest 9

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STATistics and probability

3 a Smallest total: 3, largest total: 18 1 b

15

17 No, because there is also the possibility of 1 Head (HT or TH). 18 a

1 216

4

3 216

5

6 216

6

10 216

7

15 216

8

21 216

23 a

9

25 216

d 0 e

10

27 216

24 a

11

27 216

14

15 216

15

10 216

16

6 216

17

3 216

18

1 216

1 16 2

1– 2

9 a

11 a

1– 2

1– 2

R

B

c

EV

2 1– 2

R

1– 2

B

1– 2 1– 2

R B

G

G

1– 2

1– 2

B

1– 2

G

1– 2

1– 2

1– 2

1– 2

1– 2

B G

BGB BGG

B G

GBB GBG

B G

GGB GGG

1– 8 1– 8 1– 8 1– 8 1– 8 1– 8

— 1

3 8 7 8

d They are equally likely.

e

b

3

1

1– 2

c

A

1– 2 1– 2 1– 2 1– 2 1– 2

1– 2 1– 2 1– 2

3 R B

B 2

1

R B

BRR BRB

R B

BBR BBB

8 7

26 If two events are complementary, they cannot occur at the same

time, thus their intersection is ø, the same as mutually exclusive sets. However, if events are mutually exclusive, they do not need to have a sum equal to 1. 7 a {BBBB, BBBG, BBGB, BBGG, BGBB, BGBG, BGGB, 2 BGGG, GBBB, GBBG, GBGB, GBGG, GGBB, GGBG, GGGB, GGGG} 3 8

1 2

b c 1 1– 3

R 1– 3

Outcomes Probability 1– RRR 8 1– RRB 8 RBR RBB

ξ 4

6

28 a

R B

5

3

1 b 1 2 11 19 23 5 b c d 30 30 30 6

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10 a

1– 2

B

Outcomes Probability 1– BBB 8 1– BBG 8

3 B G

1– 2

b 3

1 17

E

8 B

2

25 a S = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3}, B = {4, 5, 6}

215 4 D 216 1 3 b 4 4 17 53 9 61 b c d 70 70 70 70

2 13

B

1– 2

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7

1 1– 2

Exercise 13.6  Mutually exclusive and independent events

6 a

1 19 1 b c 17 34 221 25 26

Advantaged. The chance of getting a total of 7 would be 13. 2 A

or 0.189

1 9 1 22 a b c 16 16 169 1 2601 d e 2704 2704

11 b 221 45 17 1081



189 1000

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21 216

21

N

13

20 D

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12

25 216

19 a

Challenge 13.2

5 a

8 15 4 8 b c d 20 20 20 20 14 3 38 2 b c d 52 13 52 13

3

5 12

1 D

1 12

16

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1 221

Probability

1 8

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Total

15

1– 8 1– 8 1– 8 1– 8 1– 8 1– 8

— 1

1 3 3 8 8 8 1 7 1 iv  v vi 8 8 2 8 1 1 12 13 14 81 9 2

b   i  ii iii

1– 6

1– 2 1– 3

G

1– 2

1– 6

1– 6

1– 2 1– 3

B 1– 2

1– 6

2 Outcomes Probability 1– RR 9

R

G

RG

B

RB

R

GR

G

GG

B

GB

R

BR

G

BG

B

BB

1 — 18 1– 6 1 — 18 1 — 36 1 — 12 1 — 6 1 — 12 1– 4

— 1

b {(R, R), (R, G), (R, B)} c

1 3

d

7 18

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STATistics and probability

1– 2

1– 4

1 1– 2

1– 4 1– 2

1– 4

1– 4

2 1– 4

1– 4

1– 2

1– 4

3 1– 4

2 Outcomes Probability 1– 11 4

2

12

3

13

1

21

2

22

3

23

1

31

2

32

3

33

Die 2

5 a

1

1

1– 8 1– 8

1

1– 8 1 — 16 1 — 16 1– 8 1 — 16 1 — 16

— 1

4 (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)

1– 3

B

1– 3

1– 3

1– 3

D

E

1– 3

1– 3

ADF ADG ADH

F G

AEF AEG AEH BCF

H F –1 3 G

1– 3

1– 3

BCG BCH

H F 1– 3 G

BDF BDG BDH BEF

H

1– 3 1– 3 1– 3

F G H

1– 3

C 1– 3

F G H

BEG BEH

1 — 18 1 — 18 1 — 18 1 — 18 1 — 18 1 — 18 1 — 18 1 — 18 1 — 18 1 — 18 1 — 18 1 — 18 1 — 18 1 — 18 1 — 18 1 — 18 1 — 18 1 — 18

C 17



— 1

1 6

c

E

1 2

Exercise 13.7  Conditional probability 4 15

13 20

16 39

c i  ii

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1 a 60 b 31 16 d i  ii 60 31 4 15

4 15

b

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11

6

6

7 17 10 40 11 11 iv  v 28 17

11 40

c Order from least to most probable: iii, iv, ii, v, i. 7 62.5% 2 2 8 a 0.2 b c 9

3

5

9 35

10 a

1 1 1 b c 0 d 6 6 6

11 Conditional probability reduces the sample space that the

probability is calculated from. In this instance the sample space is reduced from 6 numbers (1, 2, 3, 4, 5, 6) to 3 numbers (2, 4, 6). 41 12 a P(J) = 90 b P(H | J) = 13 0.22 14 0.58 15 0.9

12 41

Investigation — Rich task

c They are the same, and equal to the probability calculated in question 1 part b.

Coin 1

D

b   i  ii iii

Sample space = {ACF, ACG, .  .  ., BEG, BEH}

b

O

ACG ACH

ξ

N

1– 3

D

E

ACF

F G H

1– 3 1– 3

1– 3

1– 2

1– 3

3 5

AT IO

1– 3

1– 3

1– 2

4 a

3 (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)

6 a

3 Outcomes Probability

C

A

3 10

6

(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)

AL U

2

b

EV

1

1– 3

3

5

6 (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)

1– 3

2 a

4

5 (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)

1 1 c   i  ii 4 2 3 9 iii  iv 8 16



3

2 (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)

b No

30 a

2

N LY

1

Die 1

29 a

Coin 2 Outcomes HH H

H

1 3

2 4 1 1 3 4 2 4 5 18 6 2 7 Answers will vary. 8 8 9 Answers will vary. 10 6 11 Answers will vary.

Code puzzle

T

HT

H

TH

T

TT

To remember colours of the rainbow

T

b

1 2

528  Maths Quest 9

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N LY O N AT IO AL U EV E M PL SA c13Probability.indd 529

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