CAMBRIDGE IGCSE MATHEMATICS Additional Practice Handling Data 1 Collecting and representing data 1. Monica

was doing a geography project on the weather. As part of her work, she kept a record of the daily midday temperatures in June.

Daily temperatures for June (°C) 15 20 26 18 16

a

18 23 26 18 16

19 22 20 19 17

21 24 19 17 18

23 24 19 16 20

22 25 20 15 22

Copy and complete the grouped frequency table for her data. Temperature (°C) 14–16 17–19 20–22 23–25 26–28

Tally

Frequency

b

In which interval do the most temperatures lie?

c

Describe what the weather was probably like throughout the month.

2. The

data shows the heights, in centimetres, of a sample of 32 Year 10 students.

172

158

160

175

180

167

159

180

167

166

178

184

179

156

165

166

184

175

170

165

164

172

154

186

167

172

170

181

157

165

152

164

a

Draw a grouped frequency table for the data, using class intervals 151–155, 156–160, …

b

In which interval do the most heights lie?

c

Does this agree with a survey of the students in your class? 1

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3. The

pictogram, taken from a Suntours brochure, shows the average daily hours of sunshine for five months in Tenerife.

a

Write down the average daily hours of sunshine for each month.

b

Which month had the most sunshine?

c

Give a reason why pictograms are useful in holiday brochures.

May

✹✹✹✹✹

June

✹✹✹✹✹✹

July

✹✹✹✹✹✹

August

✹✹✹✹✹✹

September Key

4. The

pictogram shows the amounts of money collected by six students after they had completed a sponsored walk for charity.

Anthony Ben Emma

a

Who raised the most money?

b

How much money was raised altogether by the six pupils?

c

Robert also took part in the walk and raised $32. Why would it be difficult to include him on the pictogram?

Leanne

✹✹✹✹✹ ✹ represents 2 hours $$$$$ $$$$$$ $$$$$ $$$$

Reena

$$$$$$

Simon

$$$$$$$

Key

$ represents $5

5. The

frequency table below shows the levels achieved by 100 Year 9 students in their KS3 mathematics tests. Level Frequency

3 12

4 22

5 24

6 25

7 15

8 2

a

Draw a suitable bar chart to illustrate the data.

b

What fraction of the students achieved Level 6 or Level 7?

c

State an advantage of drawing a bar chart rather than a pictogram for this data.

6. This

table shows the number of points Richard and Derek were each awarded in eight rounds of a general knowledge quiz. Round Richard Derek

1 7 6

2 8 7

3 7 6

4 6 9

5 8 6

a

Draw a dual bar chart to illustrate the data.

b

Comment on how well each of them did in the quiz.

6 6 8

7 9 5

8 4 6

7. Kay

did a survey on the time it took students in her form to get to school on a particular morning. She wrote down their times to the nearest minute. 15

23

36

45

8

20

34

15

27

49

10

60

5

48

30

18

21

2

12

56

49

33

17

44

50

35

46

24

11

34

a

Draw a grouped frequency table for Kay’s data, using class intervals 1–10, 11–20, …

b

Draw a bar chart to illustrate the data.

c

Comment on how far from school the students live. kamalmath.wordpress.com 2

8.

This table shows the number of accidents at a dangerous crossroads over a six-year period. Year No. of accidents

9.

2000 6

2003 9

2004 6

2005 4

Draw a pictogram for the data.

b

Draw a bar chart for the data.

c

Which diagram would you use if you were going to write to your local council to suggest that traffic lights should be installed at the crossroads? Explain why.

The table shows the estimated number of tourists world wide.

No. of tourists (millions)

1970 60

1975 100

1980 150

1985 220

1990 280

1995 290

2000 320

a

Draw a line graph for the data.

b

From your graph estimate the number of tourists in 2002.

c

In which five-year period did world tourism increase the most?

d

Describe the trend in world tourism. What reasons can you give to explain this trend?

2005 340

The table shows the maximum and minimum daily temperatures for a city over a week. Day Maximum (°C) Minimum (°C)

11.

2002 7

a

Year

10.

2001 8

Sunday 12 4

Monday 14 5

Tuesday 16 7

Wednesday 15 8

Thursday 16 7

Friday 14 4

Saturday 10 3

a

Draw line graphs on the same axes to show the maximum and minimum temperatures.

b

Find the smallest and greatest difference between the maximum and minimum temperatures.

The table shows how many students were absent from one particular class throughout the year. Students absent Frequency

1 48

2 32

3 12

4 3

5 1

Draw a frequency polygon to illustrate the data. 12.

The table shows the number of goals scored by a hockey team in one season. Goals Frequency

1 3

2 9

3 7

4 5

5 2

Draw the frequency polygon for this data. 13.

The table shows the range of heights of the girls in Y11 at a school. Height, h (cm) Frequency a

120 ⬍ h ⭐130 130 ⬍ h ⭐140 140 ⬍ h ⭐150 150 ⬍ h ⭐160 160 ⬍ h ⭐170 15 37 25 13 5

Draw a frequency polygon for this data.

b

3

Draw a histogram for this data.

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14.

Draw a pie chart to represent each of the following sets of data. a

The number of children in 40 families. No. of children Frequency

b

Frequency

2 14

3 9

4 3

Home and Away 15

Neighbours

Coronation Street 10

18

Eastenders

Emmerdale

13

4

How 90 students get to school. Journey to school Frequency

15.

1 10

The favourite soap-opera of 60 students. Programme

c

0 4

Walk 42

Car 13

Bus 25

Cycle 10

Mariam asked 24 of her friends which sport they preferred to play. Her data is shown in this frequency table. Sport Frequency

Rugby 4

Football 11

Tennis 3

Squash 1

Basketball 5

Illustrate her data on a pie chart. 16.

17.

In the run up to an election, 720 people were asked in a poll which political party they would vote for. The results are given in the table. a

Draw a pie chart to illustrate the data.

b

Why do you think pie charts are used to show this sort of information during elections?

Conservative Labour Liberal-Democrat Green Party

This pie chart shows the proportions of the different shoe sizes worn by 144 pupils in a school.

248 264 152 56

3&4 11 & 12

a

b

How many pupils had a shoe size of 11 or 12?

c

What percentage of pupils wore the modal size?

5&6

45°

What is the angle of the sector representing shoe sizes 11 and 12?

60° 120° 80°

9 & 10

7&8

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2 Measures of central tendency 1. Shopkeepers

always want to keep the most popular items in stock. Which average do you think is often known as the shopkeeper’s average?

2. The

marks of 25 students in an English examination are as follows.

55, 63, 24, 47, 60, 45, 50, 89, 39, 47, 38, 42, 69, 73, 38, 47, 53, 64, 58, 71, 41, 48, 68, 64, 75 Find the median. 3. This

table shows the annual salaries for a firm’s employees.

Chairman Managing director Floor manager Skilled worker 1 Skilled worker 2 Machinist Computer engineer Secretary Office junior

$43 $37 $25 $24 $24 $18 $18 $18 $7

000 000 000 000 000 000 000 000 000

a

What is i the modal salary,

b

The management has suggested a pay rise for all of 6%. The shopfloor workers want a pay rise for all of $1500. What difference to the mean salary would each suggestion make?

ii

the median salary, and

iii

the mean salary?

4. A

list of 9 numbers has a mean of 7.6. What number must be added to the list to give a new mean of 8?

5. The

mean age of a group of eight walkers is 42. Joanne joins the group and the mean age changes to 40. How old is Joanne?

6. Find a

i the mode, ii the median and iii the mean from each frequency table below.

A survey of the shoe sizes of all the Y10 boys in a school gave these results. Shoe size Number of pupils

b

4 12

5 30

6 34

7 35

8 23

9 8

10 3

This is a record of the number of babies born each week over one year in a small maternity unit. Number of babies Frequency

0 1

1 1

2 1

3 2

4 2

5 2

6 3

7 5

8 9

9 8

10 11 12 13 14 6 4 5 2 1

7. The

teachers in a school were asked to indicate the average number of hours they spent each day marking. The table summarises their replies. Number of hours spent marking Number of teachers

1 10

2 13

a

How many teachers are at the school?

b

What is the modal number of hours spent marking?

c

What is the mean number of hours spent marking?

5

3 12

4 8

5 6

6 1

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8. The

number of league goals scored by a football team over a season is given in the table. Number of goals scored Number of matches

0 3

1 8

2 10

3 11

4 4

5 2

6 1

7 1

a

How many games were played that season?

b

What is the modal number of goals scored?

c

What is the median number of goals scored?

d

What is the mean number of goals scored?

e

Which average do you think the team’s supporters would say is the average number of goals scored by the team that season?

f

If the team also scored 20 goals in ten cup matches that season, what was the mean number of goals the team scored throughout the whole season?

9. The

table shows the number of passengers in each of 100 taxis leaving London Airport one day. No. of passengers in a taxi No. of taxis

1 x

2 40

4 26

3 y

a

Find the value of x + y.

b

If the mean number of passengers per taxi is 2.66, show that x + 3y = 82.

c

Find the values of x and y by solving appropriate equations.

d

State the median of the number of passengers per taxi.

10.

For each table of values, find the following. i

the modal group

ii

an estimate for the mean

a

x Frequency

b

y 0 ⬍ y ⭐100 100 ⬍ y ⭐200 200 ⬍ y ⭐300 300 ⬍ y ⭐400 400 ⬍ y ⭐500 500 ⬍ y ⭐600 Frequency 95 56 32 21 9 3

c

z Frequency

d

Weeks Frequency

0 ⬍ x ⭐ 10 10 ⬍ x ⭐ 20 20 ⬍ x ⭐ 30 30 ⬍ x ⭐ 40 40 ⬍ x ⭐ 50 4 6 11 17 9

0⬍z⭐5 16 1–3 5

5 ⬍ z ⭐ 10 10 ⬍ z ⭐ 15 15 ⬍ z ⭐ 20 27 19 13 4–6 8

7–9 14

10–12 10

6

13–15 7

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11.

One hundred light bulbs were tested by their manufacturer to see whether the average life span of the manufacturer’s bulbs was over 200 hours. The table summarises the results. Life span, h (hours) Frequency

12.

150 ⬍ h ⭐175 175 ⬍ h ⭐200 200 ⬍ h ⭐225 225 ⬍ h ⭐250 250 ⬍ h ⭐275 24 45 18 10 3

a

What is the modal length of time a bulb lasts?

b

What percentage of bulbs last longer than 200 hours?

c

Estimate the mean life span of the light bulbs.

d

Do you think the test shows that the average life span is over 200 hours? Explain your answer.

The owners of a boutique did a survey to find the average age of people using the boutique. The table summarises the results. Age (years) Frequency

14–18 26

19–20 24

21–26 19

27–35 16

36–50 11

What do you think is the average age of the people using the boutique?

3 Measures of spread and cumulative frequency diagrams 1. A

class of 30 children was asked to estimate one minute. The teacher recorded the times the pupils actually said. The table on the right shows the results.

a

Copy the table and complete a cumulative frequency column.

b

Draw a cumulative frequency diagram.

c

Use your diagram to estimate the median time and the interquartile range.

group of 50 pensioners was given the same task as the children in question 1. The results are shown in the table on the right.

Time (seconds) 20 ⬍ x ⭐ 30 30 ⬍ x ⭐ 40 40 ⬍ x ⭐ 50 50 ⬍ x ⭐ 60 60 ⬍ x ⭐ 70 70 ⬍ x ⭐ 80 80 ⬍ x ⭐ 90

Number of pupils 1 3 6 12 3 3 2

Time (seconds) 10 ⬍ x ⭐ 20 20 ⬍ x ⭐ 30 30 ⬍ x ⭐ 40 40 ⬍ x ⭐ 50 50 ⬍ x ⭐ 60 60 ⬍ x ⭐ 70 70 ⬍ x ⭐ 80 80 ⬍ x ⭐ 90 90 ⬍ x ⭐ 100

Number of pensioners 1 2 2 9 17 13 3 2 1

2. A

a

Copy the table and complete a cumulative frequency column.

b

Draw a cumulative frequency diagram.

c

Use your diagram to estimate the median time and the interquartile range.

d

Which group, the children or the pensioners, would you say was better at estimating time? Give a reason for your answer.

3. The

sizes of 360 secondary schools are recorded in the table on the right.

a

Copy the table and complete a cumulative frequency column.

b

Draw a cumulative frequency diagram.

c

Use your diagram to estimate the median size of the schools and the interquartile range.

d

Schools with less than 350 pupils are threatened with closure. About how many schools are threatened with closure? 7

Number of pupils 100–199 200–299 300–399 400–499 500–599 600–699 700–799 800–899 900–999

Number of schools 12 18 33 50 63 74 64 35 11 kamalmath.wordpress.com

4. The

temperature at a seaside resort was recorded over a period of 50 days. The temperature was recorded to the nearest degree. The table on the right shows the results.

a

Copy the table and complete a cumulative frequency column.

b

Draw a cumulative frequency diagram. Note that as the temperature is to the nearest degree the top values of the groups are 7.5°C, 10.5°C, 13.5°C, 16.5°C, etc.

c

Use your diagram to estimate the median temperature and the interquartile range.

5. At

the school charity event, a game consists of throwing three darts and recording the total score. The results of the first 80 people to throw are recorded in the table on the right.

a

Draw a cumulative frequency diagram to show the data.

b

Use your diagram to estimate the median score and the interquartile range.

c

People who score over 90 get a prize. About what percentage of the people get a prize?

Temperature (°C) 5–7 8–10 11–13 14–16 17–19 20–22 23–25 26–28 29–31

Number of days 2 3 5 6 6 9 8 6 5

Total score 1 ⭐ x ⭐ 20 21 ⭐ x ⭐ 40 41 ⭐ x ⭐ 60 61 ⭐ x ⭐ 80 81 ⭐ x ⭐ 100 101 ⭐ x ⭐ 120 121 ⭐ x ⭐ 140

Number of players 9 13 23 15 11 7 2

4 Histograms – unequal class intervals 1. Draw

histograms for these grouped frequency distributions.

a

Temperature, t (°C) Frequency

8 ⭐ t ⬍10 10 ⭐ t ⬍12 12 ⭐ t ⬍15 15 ⭐ t ⬍17 17 ⭐ t ⬍20 20 ⭐ t ⬍24 5 13 18 4 3 6

b

Wage, w ($1000) Frequency

6 ⭐ w ⬍10 16

c

Age, a (nearest year) Frequency

d

Pressure, p (mm) Frequency

e

Time, t (min) Frequency

10 ⭐ w ⬍12 12 ⭐ w ⬍16 16 ⭐ w ⬍24 54 60 24

11 ⭐ a ⬍14 14 ⭐ a ⬍16 16 ⭐ a ⬍17 17 ⭐ a ⬍20 51 36 12 20

745 ⭐ p ⬍755 755 ⭐ p ⬍760 760 ⭐ p ⬍765 765 ⭐ p ⬍775 4 6 14 10

0⭐t⬍8 72

8 ⭐ t ⬍ 12 12 ⭐ t ⬍ 16 16 ⭐ t ⬍ 20 84 54 36

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2. The

following information was gathered about the weekly pocket money given to 14 year olds. Pocket money, p ($) Girls Boys

0⭐p⬍2 8 6

2⭐p⬍4 15 11

4⭐p⬍5 22 25

5⭐p⬍8 12 15

8 ⭐ p ⬍ 10 4 6

a

Represent the information about the boys on a histogram.

b

Represent both sets of data with a frequency polygon, using the same pair of axes.

c

What is the mean amount of pocket money given to each sex? Comment on your answer.

3. The

sales of the Star newspaper over 65 years are recorded in this table. Years Copies

1930–50 62 000

1951–70 68 000

1971–80 71 000

1981–90 75 000

1991–95 1995–2000 63 000 52 000

Illustrate this information on a histogram. Take the class boundaries as 1930, 1950, 1970, 1980, 1990, 1995, 2000. 4. A

city’s trains were always late, so one month a survey was undertaken to find how many trains were late, and by how many minutes (to the nearest minute). The results are illustrated by this histogram.

30

Frequency density

25 20 15 10 5 0 0

10

20

30 Minutes late

40

a

How many trains were in the survey?

b

How many trains were delayed for longer than 15 minutes?

50

60

5. One

summer, Tariq monitored the weight of the tomatoes grown on each of his plants. His results are summarised in this table. Weight, w (kg) Frequency

6 ⭐ w ⬍ 10 8

10 ⭐ w ⬍ 12 12 ⭐ w ⬍ 16 16 ⭐ w ⬍ 20 20 ⭐ w ⬍ 25 15 28 16 10

a

Draw a histogram for this distribution.

b

Estimate the median weight of tomatoes the plants produced.

c

Estimate the mean weight of tomatoes the plants produced.

d

How many plants produced more than 15 kg?

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6. A

survey was carried out to find the speeds of cars passing a particular point on a motorway. The histogram illustrates the results of the survey. 12

Frequency density

10 8 6 4 2 0 0

a

10

20

30

40

50 60 70 Speed (kmph)

80

90

100

Copy and complete this table. Speed, v (kmph) 0 ⬍ v ⭐40 40 ⬍ v ⭐50 50 ⬍ v ⭐60 60 ⬍ v ⭐70 70 ⬍ v ⭐80 80 ⬍ v ⭐100 Frequency 10 40 110

b

Find the number of cars included in the survey.

c

Work out an estimate of the median speed of the cars on this part of the motorway.

d

Work out an estimate of the mean speed of the cars on this part of the motorway.

5 Probability 1. Naseer

throws a dice and records the number of sixes that he gets after various numbers of throws. The table shows his results. Number of throws Number of sixes

10 2

50 4

100 10

200 21

500 74

1000 163

2000 329

a

Calculate the experimental probability of a six at each stage that Naseer recorded his results.

b

How many ways can a dice land?

c

How many of these ways give a six?

d

What is the theoretical probability of throwing a six with a dice?

e

If Naseer threw the dice a total of 6000 times, how many sixes would you expect him to get?

10

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2. Marie

2 27

3 32

4 53

1

3

1 19

2

Side spinner lands on Number of times

4

made a five-sided spinner, like the one shown in the diagram. She used it to play a board game with her friend Sarah. The girls thought that the spinner wasn’t very fair as it seemed to land on some numbers more than others. They spun the spinner 200 times and recorded the results. The results are shown in the table.

5

5 69

a

Work out the experimental probability of each number.

b

How many times would you expect each number to occur if the spinner is fair?

c

Do you think that the spinner is fair? Give a reason for your answer.

3. Sarah

thought she could make a much more accurate spinner. After she had made it, she tested it and recorded how many times she scored a 5. Her results are shown in the table. Number of spins Number of fives

10 3

50 12

100 32

500 107

a

Sarah made a mistake in recording the number of fives. Which number in the second row above is wrong? Give a reason for your answer.

b

These are the full results for 500 spins. Side spinner lands on Number of times

1 96

2 112

3 87

4 98

5 107

Do you think the spinner is fair? Give a reason for your answer. 4. A

sampling bottle contains 20 balls. The balls are either black or white. (A sampling bottle is a sealed bottle with a clear plastic tube at one end into which one of the balls can be tipped.) Kenny conducts an experiment to see how many black balls are in the bottle. He takes various numbers of samples and records how many of them showed a black ball. The results are shown in the table. Number of samples 10 100 200 500 1000 5000

Number of black balls 2 25 76 210 385 1987

Experimental probability

a

Copy the table and calculate the experimental probability of getting a black ball at each stage.

b

Using this information, how many black balls do you think are in the bottle?

11

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5.

A four-sided dice has faces numbered 1, 2, 3 and 4. The score is the face on which it lands. Five pupils throw the dice to see if it is biased. They each throw it a different number of times. Their results are shown in the table. Pupil

Total number of throws

Alfred Brian Caryl Deema Emma

1 7 19 102 25 61

20 50 250 80 150

Score 2 3 6 3 16 8 76 42 25 12 46 26

4 4 7 30 18 17

a

Which pupil will have the most reliable set of results? Why?

b

Add up all the score columns and work out the relative frequency of each score. Give your answers to two decimal places.

c

Is the dice biased? Explain your answer.

6.

I throw an ordinary dice 150 times. How many times can I expect to get a score of 6?

7.

I toss a coin 2000 times. How many times can I expect to get a head?

8.

I draw a card from a pack of cards and replace it. I do this 520 times. How many times would I expect to get these?

9.

a

a black card

b

a King

c

a Heart

d

the King of Hearts

The two-way table shows the age and sex of a sample of 50 pupils in a school.

Number of boys Number of girls

11 4 2

Age (years) 13 14 6 2 3 6

12 3 5

15 5 4

16 4 6

a

How many pupils are aged 13 years or less?

b

What percentage of the pupils in the table are 16?

c

A pupil from the table is selected at random. What is the probability that the pupil will be 14 years of age? Give your answer as a fraction in its lowest form.

d

There are 1000 pupils in the school. Use the table to estimate how many boys are in the school altogether.

12

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10.

The two-way table shows the number of adults and the number of cars in 50 houses in one street.

11.

1 2 3 0 0

0 1 2 3

Number of cars

Number of adults 2 3 1 0 13 3 10 6 1 4

4 0 1 4 2

a

How many houses have exactly two adults and two cars?

b

How many houses altogether have three cars?

c

What percentage of the houses have three cars?

d

What percentage of the houses with just one car have three adults living in the house?

Zoe throws a fair coin and rolls a fair dice. If the coin shows a head she records the score on the dice. If the coin shows tails she doubles the number on the dice. a

Complete the two-way table to show Zoe’s possible scores.

Coin

1 1 2

Head Tail

Number on dice 3 4

2 2 4

b

How many of the scores are square numbers?

c

What is the probability of getting a score that is a square number?

5

6

12.

Each morning I run to work or get a lift. The probability that I run to work is –52 . What is the probability that I get a lift?

13.

A letter is to be chosen at random from this set of letter-cards.

S a

an S?

S T

ii

a T?

iii

I

C S

a vowel?

picking an S / picking a T

iiipicking

14.

I

Which of these pairs of events are mutually exclusive? i

c

A T

What is the probability the letter is i

b

T

an S / picking another consonant

ii

picking an S / picking a vowel

iv

picking a vowel / picking a consonant

Which pair of mutually exclusive events in part b is also exhaustive?

At the morning break, I have the choice of coffee, tea or hot chocolate. If the probability I choose coffee is –53 , the probability I choose tea is –41 , what is the probability I choose hot chocolate?

13

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15.

A hotelier conducted a survey of guests staying at her hotel. The table shows some of the results of her survey. Type of guest Man Woman American man American woman Vegetarian Married a

Probability 0.7 0.3 0.2 0.05 0.3 0.6

A guest was chosen at random. From the table, work out these probabilities. i

the guest was American

ii

the guest was single

iiithe

guest was not a vegetarian

b

Explain why it is not possible to work out from the table the probability of a guest being a married vegetarian.

c

From the table, give two examples of pairs of types of guest that would form a pair of mutually exclusive events.

d

From the table, give one example of a pair of types of guest that would form a pair of exhaustive events.

6 The addition and multiplication rules 1.

2.

Say whether these pairs of events are mutually exclusive or not. a

tossing a head with a coin/tossing a tail with a coin

b

throwing a number less than 3 with a dice/throwing a number greater than 3 with a dice

c

drawing a Spade from a pack of cards/drawing an Ace from a pack of cards

d

drawing a Spade from a pack of cards/drawing a red card from a pack of cards

e

if two people are to be chosen from three girls and two boys: choosing two girls/choosing two boys

f

drawing a red card from a pack of cards/drawing a black card from a pack of cards

Iqbal throws an ordinary dice. What is the probability that he throws these scores? a

3.

5

c

2 or 5

a Heart

b

a Club

c

a Heart or a Club

Jasper draws a card from a pack of cards. What is the probability that he draws one of the following numbers? a

5.

b

Jennifer draws a card from a pack of cards. What is the probability that she draws these? a

4.

2

2

b

6

c

2 or 6

A letter is chosen at random from the letters on these cards. What is the probability of choosing each of these?

P R O B A B a

aB

b

I

a vowel 14

L

I

T Y c

a B or a vowel

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6. John

needs his calculator for his mathematics lesson. It is either in his pocket, bag or locker. The probability it is in his pocket is 0.35; the probability it is in his bag is 0.45. What is the probability that

a

he will have the calculator for the lesson?

b

it is in his locker?

7. Debbie

has 20 unlabelled CDs, 12 of which are rock, 5 are pop and 3 are classical. She picks a CD at random. What is the probability of these outcomes?

a

rock or pop

b

pop or classical

c

not pop First event

8. A

coin is tossed twice. Copy and complete the tree diagram below to show all the outcomes.

getting two heads

b

getting a head and a tail

c

getting at least one tail

H

1 _ 2

Use your tree diagram to work out the probability of each of these outcomes. a

Second event

Outcome

Probability

(H, H)

1 _ × 1 _ =1 _ 2 2 4

H

1 _ 2

T H

1 _ 2

T T 9. On

my way to work, I drive through two sets of road works with traffic lights which only show green or red. I know that the probability of the first set being green is –31 and the probability of the second set being green is –21 . a

What is the probability that the first set of lights will be red?

b

What is the probability that the second set of lights will be red?

c

Copy and complete the tree diagram right, showing the possible outcomes when passing through both sets of lights.

d

e

Using the tree diagram, what is the probability of each of the following outcomes? i

I do not get held up at either set of lights

ii

I get held up at exactly one set of lights

iii

I get held up at least once

First event

Second event 1 _ 2

G

Outcome

Probability

(G, G)

1 _ × 1 _ =1 _ 3 2 6

G

1 _ 3

R G R R

Over a school term I make 90 journeys to work. On how many days can I expect to get two green lights?

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10.

11.

Six out of every 10 cars in Spain are foreign made. a

What is the probability that any car will be made in Spain?

b

Two cars can be seen approaching in the distance. Draw a tree diagram to work out the probability of each of these outcomes. i

both cars will be made in Spain

ii

one car will be Spanish and the other car will be foreign made

Thomas has to take a three-part language examination paper. The first part is speaking. He has a 0.4 chance of passing this part. The second is listening. He has a 0.5 chance of passing this part. The third part is writing. He has a 0.7 chance of passing this part. Draw a tree diagram covering three events where the first event is passing or failing the speaking part of the examination, the second event is passing or failing the listening part, and the third event is passing or failing the writing part.

a

If he passes all three parts, his father will give him $20. What is the probability that he gets the money?

b

If he passes two parts only, he can resit the other part. What is the chance he will have to resit?

c

If he fails all three parts, he will be thrown off the course. What is the chance he is thrown off the course?

7 Multiplication rule for dependent events 1.

Alf tosses a coin twice. The coin is biased so it has a probability of –32 of landing on a head. What is the probability that he gets a

2.

b

a head and a tail (in any order)?

Bernice draws a card from a pack of cards, replaces it, shuffles the pack and then draws another card. What is the probability that the cards are a

3.

two heads?

both Aces?

b

an Ace and a King (in any order)?

A dice is thrown twice. What is the probability that both scores are a

even?

b

one even and one odd (in any order)?

4.

I throw a dice three times. What is the probability of getting three sixes?

5.

A bag contains 15 white beads and 10 black beads. I take out a bead at random, replace it and take out another bead. What is the probability of each of these?

6.

a

both beads are black

b

one bead is black and the other white (in any order)

The probability that I am late for work on Monday is 0.4. The probability that I am late on Tuesday is 0.2. What is the probability of each of the following outcomes? a

I am late for work on Monday and Tuesday.

b

I am late for work on Monday and on time on Tuesday.

c

I am on time on both Monday and Tuesday.

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7.

8.

I put six CDs in my multi-player and put it on random play. Each CD has 10 tracks. Once a track is played, it is not played again. a

What is the chance that track 5 on CD 6 is the first one played?

b

What is the maximum number of tracks that could be played before a track from CD 6 is played?

There are five white and one brown eggs in an egg box. Kate decides to make a two-egg omelette. She takes each egg from the box without looking at its colour. a

What is the probability that the first egg taken is brown?

b

If the first egg taken is brown, what is the probability that the second egg taken will be brown?

c

What is the probability that Kate gets an omelette made from each of these combinations? i

9.

two white eggs

ii

one white and one brown egg

iii

two brown eggs

A box contains 10 red and 15 yellow balls. One is taken out and not replaced. Another is taken out. a

If the first ball taken out is red, what is the probability that the second ball is i

b

red?

ii

yellow?

If the first ball taken out is yellow, what is the probability that the second ball is i

red?

ii

yellow?

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