28 Lesson 2-G ~ Compound Probabilities Using Lists, Tree Diagrams And Tables

COMPOUND PROBABILITIES USING LISTS, TREE DIAGRAMS AND TABLES

LESSON 2-G

Each trimester in PE a student will play one sport. For first trimester the possible sports are soccer, tennis or golf. For second trimester the possible sports are basketball or volleyball. For third trimester the possible sports are dodgeball or rugby. How many different groups of three sports could a student play? What is the probability a student will be enrolled in tennis, basketball and dodgeball if a schedule is assigned at random? Follow the steps below to answer these questions.

Step 1: One way to organize the three sports a student might play in a year is to make a list of all the possibilities. a. Two combinations are listed below. Copy and complete the list.

S, B, DS, B, R

b. How many different groups of three sports are possible for a student to be assigned in one year based on your list? How did you find the answer?

Step 2: A tree diagram is another way to organize the information to determine possible combinations. Each column in the tree diagram represents one of the trimesters. The sports are listed in the columns. Reading the chart from left to right shows the different possible groups of sports a student may play. a. Copy and complete the tree diagram below.

S

B

V

D

R

D

R

T

BD

R

G

EXPLORE! THREE SPORTS

Lesson 2-G ~ Compound Probabilities Using Lists, Tree Diagrams And Tables 29

Step2: b. How many different groups of three sports are possible for a student to be assigned in one year based on your tree diagram? How did you find the answer?

Step 3: A table is a third way to organize the information. The first column shows the possible sports first trimester. The first row show the possible sports second trimester. Inside the chart are the possible sports third trimester. This shows the different possible groups of sports a student may play.

a. Copy and complete the table below.

B V

S D R

T

G

b. How many different groups of three sports are possible for a student to be assigned in one year based on your table? How did you find the answer?

Step 4: Assume students are randomly scheduled into a sport each trimester. What is the probability a student will be enrolled in tennis, basketball and dodgeball? P(tennis, basketball, dodgeball) = number of times “tennis, basketball, dodgeball” listed

_____________________________________ total number of different groups of sports listed =

Step 5: In this Explore! you listed the possible outcomes three different ways – using a list, a tree diagram and a chart. Which one did you like best for organizing the information and counting the possible outcomes? Explain.

Listing the possible outcomes shows a sample space. In the Explore! you listed the possible outcomes for sports that could be played by a person in PE for one year. It is important to be able to see and count the number of possible outcomes in a sample space to find probabilities.

EXPLORE! (CONTINUED)

2nd Trimester1s

t Trim

este

r

30 Lesson 2-G ~ Compound Probabilities Using Lists, Tree Diagrams And Tables

You are packing for a trip. You decide to take four shirts (red, blue, green and yellow) and three shorts (black, brown and plaid). How many outfits are possible?

Choose one of the methods below to organize the information and see all the possible outfits.

Tree Diagram List Table

redblack

brownplaid

blueblack

brownplaid

greenblack

brownplaid

yellowblack

brownplaid

Shirt Shorts Shirt, Shorts

red, blackred, brownred, plaid

blue, blackblue, brownblue, plaid

green, blackgreen, browngreen, plaidyellow, black

yellow, brownyellow, plaid

Black Brown Plaid

Red RedBlack

Red Brown

RedPlaid

Blue BlueBlack

BlueBrown

BluePlaid

Green GreenBlack

GreenBrown

GreenPlaid

Yellow YellowBlack

YellowBrown

YellowPlaid

There are 12 different outfits possible.

Rolling a number cube and tossing a coin are two separate events. You can make a list to show the 12 possible outcomes.

1, Heads 3, Heads 5, Heads1, Tails 3, Tails 5, Tails2, Heads 4, Heads 6, Heads2, Tails 4, Tails 6, Tails

What if you wanted to find the probability of rolling an odd number and the coin landing tails? This would be an example of finding a compound probability.

A compound probability is the probability of two or more events occurring. Sometimes the events are independent, which means one does not affect the other. Rolling a number cube and tossing a coin are independent events. Sometimes the events are dependent events, which means one event depends on the other event. Choosing one card from a deck of cards, keeping it, and then choosing a second card is an example of dependent events. By keeping the first card you have changed the possible cards to choose from the second time.

To find a compound probability make a list, tree diagram or table to count the number of possible outcomes (sample space) and count the number of favorable outcomes (events).

There were 12 possible outcomes for rolling a number cube and tossing a coin. Three of them showed rolling an odd number and tossing tails (1, Tails; 3, Tails; 5, Tails).

P(rolling odd number, Tails) = 3 112 4

=

EXAMPLE 1

solution

Lesson 2-G ~ Compound Probabilities Using Lists, Tree Diagrams And Tables 31

Cindy has three letter cards that spell out the word CAP. Cindy picks three cards, one at a time, without replacing them. What is the probability that she spells the word CAP in the correct order?

Create a tree diagram showing the possible combinations.

Count the number of possible outcomes. 6 total outcomes

Count the number of favorable outcomes. C, A, P → 1 favorable outcome

Find the probability. P(C, A, P) = 16

LaSean spins the spinner at the right two times. Find the probability that he spins a 3 and then a number greater than 1.

Organize the information by making 1, 1 2, 1 3, 1 4, 11, 2 2, 2 3, 2 4, 21, 3 2, 3 3, 3 4, 31, 4 2, 4 3, 4 4, 4

a list. The spins are listed in order: first spin, second spin.

Count the number of possible outcomes There are 16 possible outcomes in the list.in the sample space.

Count the number of favorable outcomes 3, 2 3, 3 3, 4in the sample space. There are three favorable outcomes.

Find the probability. P(3, number > 1) = 3 __ 16

EXAMPLE 2

solution

EXAMPLE 3

solution

1 234

C

A

P

A

C

P

P

C

A

P

A

P

C

A

C

32 Lesson 2-G ~ Compound Probabilities Using Lists, Tree Diagrams And Tables

EXERCISES

Use a list to organize the possible outcomes for each experiment. Write the number of possible outcomes listed. 1. Roll a number cube and toss a coin. 2. Spin two different spinners.

ABC

Spinner 2Spinner 1

1 234

3. Toss two coins. 4. Toss three coins.

Use a tree diagram to organize the possible outcomes for each experiment. Write the number of possible outcomes shown. 5. Choose between three hats (baseball, visor or knit) and two pairs of shoes (cleats, boots).

6. Choose one cone (sugar or waffle), one scoop of ice cream (vanilla, chocolate or strawberry) and one topping (cherry or syrup).

7. Pick one card (1, 2 or 3), toss a coin and roll a number cube.

Use a table to organize the possible outcomes for each experiment. Write the number of possible outcomes shown. 8. Roll two number cubes 9. Spin two different spinners.

ABC

Spinner 2Spinner 1

1 234

Find the number of possible outcomes for each situation using a list, tree diagram or table. Show your work.

10. You are buying T-shirts for your sports team. The shirts come in three colors (blue, red, or white) and can be either short sleeved or long sleeved. How many T-shirts are possible?

11. Jar A contains two marbles (green and white). Jar B contains three marbles (blue, yellow and red). Jar C contains two marbles (purple and pink). How many outcomes are possible if you choose one marble from each jar?

Lesson 2-G ~ Compound Probabilities Using Lists, Tree Diagrams And Tables 33

12. At a restaurant you can choose one entrée (beef or chicken), one side dish (potatoes, green beans, rice or french fries) and one dessert (cake or ice cream). How many dinners are possible?

13. Pedro is leaving on a trip. He packs three shirts (blue, green and black), two pairs of shorts (brown and navy) and three pairs of shoes (tennis, sandals and flip-flops). How many different outfits are possible?

14. You toss two coins and roll a number cube. How many outcomes are possible?

Find each probability. Use a list, tree diagram or table to identify the favorable outcomes and the sample space.

15. Find the probability of rolling two 1s in a row using a number cube.

16. Toss three coins. Find the probability of tossing exactly two heads.

17. Toss three coins. Find the probability of tossing at least two heads.

18. Tabitha has a deck of cards numbered 1-10. She picks one card, puts it back in the deck and then chooses a second card. What is the probability that she picks an even number and then a 3?

19. Javier has a deck of cards numbered 1-10 and a number cube. He chooses one card and rolls the number cube. What is the probability that he picks a number divisible by 5 and rolls a 5?

20. Tim has cards with the letters E, B, K, I on them. He picks one card, keeps it and then picks the next card until all cards are chosen. What is the probability Tim picks cards in the order B, I, K, E?

21. Kim has three jars holding marbles. The first jar has a red, green and yellow marble. The second jar has a blue and white marble. The third jar has a pink, black and brown marble. Kim picks a marble from each jar without looking. What is the probability she has a red, a white and a pink marble in her hand?

22. Dylan has a bag with 4 green marbles and 2 blue marbles. She takes one marble out of the bag and sets it on the table. Then, without replacing the marble, she chooses a second marble. What is the probability she chooses a green and a blue marble in any order?

23. Trent is making a sandwich. He has two breads to choose from (white or wheat), three meats to choose from (turkey, roast beef or ham), two vegetables to choose from (tomato or lettuce) and two condiments to choose from (mustard or mayonnaise). Trent randomly picks one bread, one meat, one vegetable and one condiment. What is the probability his sandwich is turkey and lettuce on wheat bread with mustard?