COMPOUND PROBABILITIES USING MULTIPLICATION & SIMULATION

LESSON 2-H

M

aya was making sugar cookies. She decorated them with one of two types of frosting (white or pink), one of three types of sprinkles (chocolate, rainbow or green) and one of two types of candy (peppermint or caramel). The tree diagram and list show the possible outcomes for the types of cookies Maya made. Chocolate White

Rainbow Green

Peppermint Caramel

Possible Outcomes

Peppermint

White, Chocolate, Peppermint White, Chocolate, Caramel White, Rainbow, Peppermint White, Rainbow, Caramel White, Green, Peppermint White, Green, Caramel Pink, Chocolate, Peppermint Pink, Chocolate, Caramel Pink, Rainbow, Peppermint Pink, Rainbow, Caramel Pink, Green, Peppermint Pink, Green, Caramel

Caramel Peppermint Caramel Peppermint

Chocolate Pink

Rainbow Green

Caramel Peppermint Caramel Peppermint Caramel



Maya made a total of 12 different cookies using her ingredients. Notice that the total number of outcomes is the product of the number of frosting options, sprinkle options and candy options (2 ∙ 3 ∙ 2 = 12). The Multiplication Counting Principle relates the number of choices to the number of outcomes. This principle helps identify the number of outcomes without having to show the possible outcomes in a list, tree diagram or table.

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Lesson 2-H ~ Compound Probabilities Using Multiplication & Simulation

EXAMPLE 1

Ice cream sundaes at Gary’s Creamery come in five flavors with four possible toppings. How many different sundaes can be made with one flavor of ice cream and one topping?

Solution

Multiply the number of options for ice cream with the number of options for toppings.

4 ∙ 5 = 20

There are a total of 20 possible sundaes. EXAMPLE 2

Oregon issues license plates consisting of three letters and three numbers. There are 26 letters and the letters may be repeated. There are ten digits and the digits may be repeated. How many possible license plates can be issued with three letters followed by three numbers?

Solution

The license plate has six total letters and numbers. The first three are letters (A-Z) followed by three numbers (0 – 9). Multiply the possibilities.

26 ∙ 26 ∙ 26 ∙ 10 ∙ 10 ∙ 10 = 17,576,000

There are a total of 17,576,000 license plate options. EXAMPLE 3

There are five students running a race. How many possible ways can they finish first, second and third?

Solution

There are five students to choose from for first place. There will then only be four left to choose from for second place and three left to choose from for third place. Multiply the possibilities.

5 ∙ 4 ∙ 3 = 60

There are a total of 60 different ways the students could finish first, second and third.

When finding compound probabilities you must know the number of favorable outcomes and the number of possible outcomes in the sample space. You can use a list, tree diagram, table or the Multiplication Counting Principle to determine the number of favorable outcomes and possible outcomes.

Lesson 2-H ~ Compound Probabilities Using Multiplication & Simulation 

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EXAMPLE 4

Ross has a bag of marbles that has three red, four blue and five green marbles. He chooses one marble, replaces it and then chooses a second marble. What is the probability he chose a red marble and then a green marble?

Solution

Multiply to find the total number of outcomes possible. There are 12 marbles to choose from each draw.

12 ∙ 12 = 144

Multiply to find the number of favorable outcomes. There are 3 possible red marbles to choose from in the first draw and 5 possible green marbles to choose from in the second draw.

3 ∙ 5 = 15

Find the probability.

P(red then green) =

15 5 = 144 48

The probability Ross chose a red marble and then a green marble is

EXAMPLE 5

A multiple choice test has five questions. Each question has four options to choose from. Marty randomly guesses on every problem. What is the probability he guessed correctly on each problem?

Solution

Multiply to find the total number of outcomes possible. There are 4 choices on each of the 5 questions.

4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 = 1024

Multiply to find the number of favorable outcomes. There is 1 correct answer for each of the 5 questions.

1∙1∙1∙1∙1=1

5 . 48

1 Find the probability. P(guess correctly) = ____ ​ 1024    ​ ≈ 0.00098

The probability Marty guessed correctly on all the questions is about 0.00098 or 0.098%.

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Lesson 2-H ~ Compound Probabilities Using Multiplication & Simulation

Sometimes it is difficult to make a list, tree diagram or table to show all the possible outcomes. Other times the events depend on one another and so the Multiplication Counting Principle cannot give the number of outcomes in the sample space or the number of favorable outcomes. Simulations can give a good estimate for a probability when it is difficult to determine. A simulation is an experiment you use to model a situation. You can use coins, number cubes, random number generators or other objects to simulate events. The more trials you simulate, the better your estimate for a probability. The Explore! shows a simulation. EXPLORE!

PROBABILITY SIMULATION

Ainsley has a bag of marbles that has three red, four blue and five green marbles. She chooses one marble, does not replace it and then chooses a second marble. What is the probability she chose a red marble and then a green marble? Step 1:  To answer this question, you will perform a simulation. Place three red, four blue and five green marbles in a bag or box. Copy the frequency table below onto your paper. Red then Green? Yes

No

Step 2: Without looking, choose a marble from the bag or box. Place it on your desk. Then choose a second marble from the bag or box and set it on your desk. Make a tally in your frequency table under ‘Yes’ if  you picked a red marble and then a green marble. Make a tally under ‘No’ if you did not pick a   red marble and then a green marble. Return the marbles to the bag. Step 3: Continue the simulation until you have 10 results (tally marks). You have finished 10 trials. Compute the experimental probability below. Write the probability as a percent. frequency of red and then green total number of trials

P(Red then Green) = ______________________ ​          ​

Step 4: Continue the simulation 40 more times (total of 50 trials). Make a tally in the table for each result. Compute the experimental probability again. Write the probability as a percent. frequency of red and then green total number of trials

P(Red then Green) = ______________________ ​          ​

Step 5: Compare the experimental probabilities within the class. What is your estimate for the probability Ainsley chooses a red marble and then a green marble? Why? Step 6: The theoretical probability that Ainsley will choose a red marble and then a green marble is  approximately 11%. How close is your estimated probability from the experiment? What might  make your estimate be closer to the theoretical probability for choosing red then green?



Lesson 2-H ~ Compound Probabilities Using Multiplication & Simulation 

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EXERCISES Find the number of possible outcomes for each situation.

1. A soccer team’s kit consists of two jerseys, two pairs of shorts and two pairs of socks. How many soccer

outfit combinations are possible if each outfit contains one jersey, one pair of shorts and one pair of socks?

2. Heather narrowed her clothing choices for the big party down to three skirts, two tops and four pairs of shoes. How many different outfits are possible from these choices?

3. The ice cream shop offers 31 flavors. You order a double-scoop cone. If you want two different flavors, how many different ways can the clerk put the ice cream on the cone?

4. The roller skating store sells girls’ roller skates with the following options: Colors: white, beige, pink, yellow, blue Sizes: 4, 5, 6, 7, 8 Extras: tassels, striped laces, bells Assume all skates are sold with ONE extra. How many possible arrangements exist?

5. A pizza shop offers 10-inch, 12-inch and 16-inch sizes with thin, thick,

deep dish or garlic crust. Also, the customer can choose one topping from extra cheese, pepperoni, sausage, mushroom and green pepper. How many pizza combinations are possible?

6. How many ways can six people stand in line at the movies? 7. One coin is tossed three times. How many outcomes are possible? 8. A phone number has seven numbers and starts with a 3-digit area code.

However, the 7-digit number cannot start with 0 (that calls the operator). a. How many different 7-digit phone numbers are possible in each area code? b. Why do some areas have more than one area code?



9. There are fifteen school bands participating in a competition. In how many ways can first, second and third place be awarded?

Find each probability.

10. Four coins are tossed. What is the probability of tossing four heads? 11. In a school lottery, each person chooses a 3-digit number using any of the numbers 0 – 9 for each digit.

One 3-digit number is chosen from all possible 3-digit numbers. What is the probability of winning the school lottery?

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Lesson 2-H ~ Compound Probabilities Using Multiplication & Simulation

12. A bag contains 10 red marbles, 3 green marbles and 2 white marbles. Ed chooses one marble,

replaces it and then chooses another marble. a. What is the probability he will choose two red marbles? b. What is the probability he will choose a red marble and then a green marble? c. What is the probability he will choose a green marble and then a white marble?

13. You roll a number cube three times. What is the probability of rolling a five every time? 14. Megan picks a card from a deck of cards numbered 1 – 10 and rolls a number cube. What is the probability she chooses a card with a 5 and rolls an even number?

15. Natasha is given a four-digit password for her ATM account. Every

ATM password uses the digits 0 – 9 which can be repeated in the password. What is the probability Natasha’s password is 1234?

16. Justin is given a password for his ATM account. It is a four-digit

password using the digits 0 – 9 and the digits cannot be repeated. What is the probability Justin’s password is 1234?

17. An ice cream comes in either a cup or a cone and the flavors available are chocolate, strawberry and

vanilla. If you are given an ice cream at random, what is the probability it will be a cup of chocolate ice cream?

18. What is the probability that you roll a number divisible by 3 on a number cube twice in a row? 19. You have cards with the letters C, S, M, I, U on them.

a. You pick one card, keep it and then pick the next card. This is repeated until all the cards are chosen.  What is the probability you pick the cards in the order M, U, S, I, C? b. You pick one card, keep it and then pick the next card. This is repeated until all the cards are chosen. What is the probability the first three cards are S, U, M, in that order? c. You pick one card, replace it and then pick the next card. This is repeated until five cards are picked.  What is the probability you pick the cards in the order M, U, S, I, C? d. You pick one card, replace it and then pick the next card. This is repeated until five cards are picked. What is the probability the first three cards are S, U, M, in that order? e. Do you have a better chance of picking the cards in the order M, U, S, I, C if you keep the cards or replace them after each pick? Explain your answer.

20. Twenty-five percent of the jelly beans in a jar are orange. Yellow jelly

beans make up one-fifth of the total. Five percent are white. The other half of the jar contains blue and green jelly beans. a. What is the probability of picking an orange jelly bean, replacing it and  then picking a white jelly bean? b. What is the probability of picking a yellow, then orange, then finally a green or blue jelly bean if you replace the jelly beans after each pick?



Lesson 2-H ~ Compound Probabilities Using Multiplication & Simulation 

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Perform each simulation to estimate the probability.

21. A new ice skating rink opened. The owner gave each person a red, blue or green glow bracelet. Suppose



there is an equal chance of getting any color of glow bracelet. a. What is the theoretical probability of getting a blue glow bracelet? b. Simulate this probability using a number cube. Let the numbers 1 and 2 represent a red glow  bracelet. Let the numbers 3 and 4 represent a blue glow bracelet. Let the numbers 5 and 6 represent  a green glow bracelet. Roll the number cube 50 times and place a tally under each “color” rolled in  each trial. Red

Blue

Green

1 or 2

3 or 4

5 or 6

c. What is your experimental probability of getting a blue glow bracelet? How close is your experimental probability to the theoretical probability in part a? d.  Suppose you actually have a 1 chance of getting a red glow bracelet, a 1 chance of getting a blue 3 6  glow bracelet and a 1 chance of getting a green glow bracelet. How would the simulation need to 2  change to reflect these probabilities? Explain your answer.



22. There is a 20% chance a person exposed to a virus will become sick. If you are exposed to the virus three times, what is the probability you will become sick? Follow Steps 1-3 to simulate this situation. Step 1: Use a random number generator on a calculator or place ten pieces of paper (numbered 1–10) in a bag to randomly pull. Step 2: Let the numbers 1 and 2 represent a person becoming sick. Step 3: Show three random numbers from a calculator generator or pull three numbers from a hat, replacing the number after each pick. If one of the numbers is a 1 or 2 you became sick. Put a tally in the ‘Sick’ column. If all the numbers are 3–10 place a tally in the ‘Not Sick’ column.

Sick



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Not Sick

a. Complete 50 trials (three pulls each) and record your tallies. What is the probability you will become  sick if you are exposed to the virus three times? b. How would this simulation need to change if there was a 60% chance of getting sick after being  exposed to the virus? c. How would this simulation need to change if there was a 75% chance of getting sick after being  exposed to the virus?

Lesson 2-H ~ Compound Probabilities Using Multiplication & Simulation