COMPOUND PROBABILITIES USING LISTS, TREE DIAGRAMS …mrvsmath.weebly.com/uploads/1/3/4/5/13456042/ccss_stage_2_booklet_student_edition.pdfLesson 2-G ~ Compound Probabilities Using

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


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


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

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EXAMPLE 3

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1 234

C

A

P

A

C

P

P

C

A

P

A

P

C

A

C


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?


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?

34 Lesson 2-H ~ Compound Probabilities Using Multiplication & Simulation

COMPOUND PROBABILITIES USING MULTIPLICATION & SIMULATION

LESSON 2-H

Maya 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.

White

Pink

Chocolate

Rainbow

Green

Chocolate

Rainbow

Green

Peppermint

Caramel

Peppermint

Caramel

Peppermint

Caramel

Peppermint

CaramelPeppermint

CaramelPeppermint

Caramel

Possible Outcomes

White, Chocolate, PeppermintWhite, Chocolate, Caramel

White, Rainbow, PeppermintWhite, Rainbow, CaramelWhite, Green, Peppermint

White, Green, CaramelPink, Chocolate, Peppermint

Pink, Chocolate, CaramelPink, Rainbow, Peppermint

Pink, Rainbow, CaramelPink, Green, Peppermint

Pink, Green, 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.

Lesson 2-H ~ Compound Probabilities Using Multiplication & Simulation 35

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?

Multiply the number of options for 4 ∙ 5 = 20ice cream with the number of optionsfor toppings.

There are a total of 20 possible sundaes.

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?

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.

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

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.

EXAMPLE 1

solution

EXAMPLE 2

solution

EXAMPLE 3

solution


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?

Multiply to find the total number of 12 ∙ 12 = 144outcomes possible. There are 12 marblesto choose from each draw.

Multiply to find the number of favorable 3 ∙ 5 = 15outcomes. There are 3 possible red marbles to choose from in the first draw and 5 possible green marbles to choosefrom in the second draw.

Find the probability. P(red then green) = 15 5144 48

=

The probability Ross chose a red marble and then a green marble is 548

.

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?

Multiply to find the total number of 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 = 1024outcomes possible. There are 4 choices on each of the 5 questions.

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

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

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

EXAMPLE 5

solution

EXAMPLE 4

solution


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.

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.

P(Red then Green) = frequency of red and then green ______________________ total number of trials

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.

P(Red then Green) = frequency of red and then green ______________________ total number of trials

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?

EXPLORE! PROBABILITY SIMULATION


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?


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?


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 Green1 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

3 chance of getting a red glow bracelet, a 1

6 chance of getting a blue

glow bracelet and a 12

chance of getting a green glow bracelet. How would the simulation need to 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 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-I ~ Random Sampling 41

RANDOM SAMPLING

LESSON 2-I

Sometimes people want to know something about a large group of people and they cannot ask each person. For example, if Kira is running for seventh grade class president, she may want to know her chances of winning the election. Suppose her school has 800 seventh graders. Asking each person whether they will vote for Kira will take too much time. Instead, Kira needs to find a way to ask a smaller number of seventh graders and then use that information to predict whether or not she has a chance to win. But who should Kira survey in order to get an accurate idea of who will win the election when all the seventh graders vote?

Kira surveyed each of the following groups to find the percent of each group that would vote for her in the election.

40 seventh grade girls80% vote for Kira

40 seventh grade boys35% vote for Kira

40 students in Kira’s Math Class

75% vote for Kira

40 random seventh graders65% vote for Kira

Which group of students gives Kira the most accurate prediction?

Every time Kira asked 40 students who they would vote for, she was asking a sample of seventh graders instead of all the seventh graders. A sample is a smaller group that is used to make conclusions about the entire population. The population is the entire group. In the example above, the population is the group of 800 seventh graders that attend Kira’s school. The samples are each a group of 40 students that Kira surveyed.

42 Lesson 2-I ~ Random Sampling

The most accurate information that Kira will want to use in making her decision about running for class president will come from the sample that most closely resembles the entire seventh grade class. This is the representative sample. Kira needs to choose the best representative sample for an accurate prediction.

40 seventh grade girls80% vote for Kira

40 seventh grade boys35% vote for Kira

40 students in Kira’s Math Class

75% vote for Kira

40 random seventh graders65% vote for Kira

When Kira pulled 40 names from a hat, she selected a random sample of students. A random sample is created when every member of the population is equally likely to be chosen as part of the sample. Since the names of all 800 seventh graders were in the hat, each seventh grader had an equal chance of having his or her name drawn from the hat.

It is important to have a large enough sample that will be representative of the population. If Kira had only asked three people, she would not be able to make an accurate prediction. The more people she asks, the more accurate her prediction will be.

Sometimes samples are biased. A biased sample is created when people in a sample do not accurately represent the entire population. When Kira chose to ask only girls she created a biased sample. Both boys and girls will vote for class president so both need to be included in the sample. Even though it is possible for a random sample to have bias, it is not likely.

Margo wanted to know if people in her city would vote for a dog park. She could not ask everyone in the city so she decided she would call 50 random people from the list of people who take their animals to the vet.

Explain whether or not the sample of people surveyed will most likely give an accurate prediction of how people will vote in the election.

This sample will most likely NOT give an accurate prediction. People who go to the vet own animals and may be more likely to support a vote for a dog park. The sample needs to include a better representation from the entire city, both people who own animals and people who do not. It is a biased sample.

EXAMPLE 1

solution


A company is making a new video game. There is a possibility some of the games have a defect and do not work. The owner of the company wants to make sure the games work before he sells them. He has made 10,000 games and packed them into 100 boxes that each holds 100 games. He is trying to decide if he should test 100 games from a single box or if he should test 100 games by testing one game in each of the 100 different boxes.

Which sample do you think would give a more accurate prediction about the games and the possible defect?

Choosing a single game from each of 100 different boxes would most likely give a more accurate prediction of whether or not the games have a defect. Games from the same box are usually manufactured and packed at the same time. Games from different boxes were most likely made at different times and will better represent the entire population of games.

Step 1: Write down your favorite color of the four listed below on a piece of paper. This will be used to create samples and make a prediction about the most popular color in the class. RED BLACK GREEN BLUE

Step 2: Copy the chart to use for each sample in Steps 3-6.

Sample Red Black Green Blue Prediction for most popular color in the whole class

All boysAll girls

5 students15 students

Step 3: Ask the boys to stand and share their colors chosen. Fill in the table. Make a prediction.

Step 4: Ask the girls to stand and share their colors chosen. Fill in the table. Make a prediction.

Step 5: Pull 5 names from a hat and ask the students to stand and share their colors chosen. Fill in the table. Make a prediction.

Step 6: Pull 15 names from a hat and ask the students to stand and share their colors chosen. Fill in the table. Make a prediction.

Step 7: Count the number of colors chosen by the students in the entire class. Which color did the majority of students choose? Which sample predicted that color? Explain why a sample may or may not have predicted the color actually chosen by most of the students.

EXPLORE! FAVORITE COLOR

EXAMPLE 2

solution

44 Lesson 2-I ~ Random Sampling

EXERCISES

1. What is the difference between a sample and a population?

2. Will a random sample always accurately predict the outcome for the entire population? Why or why not?

3. How can you guarantee a prediction from a survey is accurate?

Identify each type of sample as representative or biased. Explain your reasoning in a sentence.

4. A survey about a “favorite cereal” is given to every fifth person who enters a supermarket.

5. Paul asks all his friends to identify their favorite item on the school lunch menu.

6. A movie studio surveys all the adults in an audience leaving a children’s movie to “rate the film”.

7. Jill randomly picks 100 people from a phone directory and asks them about their “favorite restaurant”.

Consider the given information and identify the best sample. Explain your reasoning.

8. David wants to know which football team is the most popular in the country. Which sample will give him the best chance to make an accurate prediction? A. A survey of 80 men walking into the stadium before the game. B. A survey of 80 men and women leaving the stadium after the game. C. A survey of 80 people randomly selected from a sporting goods company’s mailing list. D. A survey of 80 people randomly selected from a national phone directory.

9. Olga has just moved to a new town and she needs to find a dentist. Olga wants to choose the best dentist in town. What is the best way for Olga to determine her dentist? A. Post a survey on a website for a week. B. Distribute a survey to patients leaving their dentist appointments at one dentist location. C. Question every fourth person entering the local supermarket about their personal experience with their own dentist.

D. Ask a few of her new neighbors who they think is the best dentist in town.

Each sample used in a survey below is biased. Explain how to modify the sample to eliminate the bias.

10. Joe wants to know what percent of people are likely to go see the new comedy film that is showing at the local theater. He asks everyone that he finds renting comedies at the movie rental store if they are planning to go see the new movie.


11. David wants to determine which of the four pizza restaurants in his town is the most popular. He stands outside each restaurant and counts people coming and going for one hour. Since he has to move to four different

restaurants, David visits one restaurant at 1:00 pm, another restaurant at 3:00 pm, the third restaurant at 5:00 pm and the last restaurant at 7:00 pm. 12. Jose wants his dad to raise his weekly allowance. To determine how much he should ask for, Jose surveys all his friends to learn what their weekly allowance is.

Is each prediction as accurate as possible based on the sample surveyed? Explain why or why not.

13. A farmer wants to know if his trees have been infected with a virus. He tests two trees out of one thousand and neither of his trees is sick. He decides all of his trees must be fine.

14. Ms. Smie wants to know if her students did their homework. She asks all of the students in her homework club if they finished. Nearly all of the students in the homework club finished, so she assumes most of her students will have finished their homework.

15. Tom wants to know which candidate people in his town will choose for mayor. He surveys 50 people who live on his street. They all choose Karen. Tom predicts Karen will win the election.

16. Storm wants to know how many students at his school have cell phones. There are 200 students in his school. He asks every fourth student who enters the building until he reaches 50 students. Based on their answers, he determines about 70% of the students at his school have cell phones.

PIZZA

46 Lesson 2-J ~ Inferences About A Population

INFERENCES ABOUT A POPULATION

LESSON 2-J

George wanted to know if he needed to spray the plants on his farm to treat a virus. He tested several plants in each of his eight fields to create a random sample. George learned that none of the plants he tested had the virus. It is very expensive to spray his fields and protect his plants from the virus. Should George buy the spray?

George decided to use the information from his random sample of plants to make a conclusion. Since none of the plants in his sample had the virus, George decided that his fields were safe from the virus. He chose not to spray his fields. Using the sample to make a conclusion does not guarantee all his plants are safe from the virus. It does mean his plants are likely safe.

An inference is a logical conclusion based on known information. In statistics, people use data from a sample to make an inference, or prediction, about an entire population.

For example, a presidential candidate may use random sampling to poll a group of people by asking who they will vote for in a specific election. Based on the information the candidate learns from the poll, he or she may determine their chance of winning the election. Because the candidate does not know if he or she will win for sure, the prediction made from the sample is an inference.

Tasha needs to create a menu for a dinner party. There will be 300 people eating dinner. She has to decide whether she should serve chicken or steak and wants input from the people attending. She asked a random sample of people coming to the party which they would prefer. The results are below.

Chicken Steak40 10

What do you think Tasha should serve? Why?

Find the total number of people surveyed. 40 + 10 = 50

Determine the percent wanting each type Chicken = 4050

= 80%of meal.

Steak = 1050

= 20%

Make an inference. Tasha should serve chicken because her survey suggests most guests may prefer chicken.

EXAMPLE 1

solution

Lesson 2-J ~ Inferences About A Population 47

Sometimes people use more than one random sample to make an inference. Suppose Tasha’s sample showed 25 people chose chicken and 25 people chose steak. It would be difficult to make an accurate inference about the entire population based on that sample. Tasha would need to sample another group from the population before making her decision.

Also, sometimes a random sample can be biased without you knowing it is. To avoid making an incorrect conclusion, some people use multiple random samples to make an inference. Multiple random samples can show the possible variations in a conclusion.

Mr. King asked two groups of 100 students in the seventh grade if they would prefer to read the information in their class from a tablet, computer or textbook. The data collected is shown below. Make at least two inferences based on the results.

Tablet Computer Textbook TotalSample 1 60 30 10 100Sample 2 69 16 15 100

Find the total number of people surveyed. 100 in each sample

Determine the percent choosing each Tablet Computer TextbookSample 1

%60 60

100= %

30 30100

= %10 10

100= %

Sample 2%

69 69100

= %16 16

100= %

15 15100

= %

reading source.

Make inferences. Two example inferences are listed below.1. Most seventh grade students prefer reading from a tablet.2. Seventh grade students prefer reading from a textbook the least.

In Example 2 the first inference is stronger than the second. There are significantly more students in each sample group that chose a tablet over both a computer and a textbook so the chance of that inference being accurate is high. Since the two samples match, the conclusion that most seventh grade students prefer reading from a tablet is strong.

However, in the second sample group, the number of students who chose a computer and the number of students who chose a textbook were nearly equal. This data was different from the first sample group where clearly more students chose a computer over a textbook. Based on these two samples it would be difficult to make an accurate conclusion about all seventh graders choosing between a computer and a textbook. But since the lowest number of students in both groups chose a textbook, we can infer that fewer seventh graders in the overall population will choose a textbook. This inference is weak, especially compared to the first inference. Selecting another sample of seventh graders might be a good idea in order to improve the conclusion about textbooks.

Sometimes it is helpful to combine the results from multiple samples to make an inference rather than look at each sample. Combining the samples from Example 2 is shown in the chart below. These numbers also support the two inferences made in the solution in Example 2.

Tablet Computer Textbook TotalSample 1 & 2 129 46 25 200

Sample 1 & 2%

129 64.5200

= %46 23

200= %

25 12.5200

= %

EXAMPLE 2

solution

48 Lesson 2-J ~ Inferences About A Population

EXERCISES

1. Tami asked a random group of 80 dog owners in her neighborhood if they used the dog park with their pets. Of those asked, 60 had taken their pets to the dog park in the last week. Tami inferred that most dog owners in her neighborhood use the dog park. Is this inference strong or weak? Explain.

2. Jeremy asked a random group of 50 students at his school if they used mechanical pencils. There are 1,200 students at his school. Of those asked, 20 said they used mechanical pencils. Jeremy inferred that students don’t use mechanical pencils. Is this inference strong or weak? Explain.

3. Laramie wanted a cell phone. He decided to ask students at his school whether or not they had cell phones. There are 800 students at Laramie’s school. He asked every fifth student who came to school until he reached a total of 100 students if they had a cell phone. Sixty students had cell phones. He inferred that almost all students at his school had a cell phone. Change his inference to make it more accurate.

Use the information provided to make an inference. Identify whether your inference is strong or weak and explain why.

4. Eighty out of 404 seventh grade boys were surveyed regarding their hair color.

Color of Hair for Seventh Grade Boys Brown Black Blonde Red TOTAL

Sample of boys 31 24 22 3 80

5. One hundred-twenty seventh grade girls were asked, “What is your favorite sport to play?” There were a total of 432 seventh graders at the school.

0

10

20

30

40

50

60

Volleyball Softball Soccer Basketball

Num

ber o

f 7th

gra

de g

irls

Favorite Sport

12 18 48 42

Lesson 2-J ~ Inferences About A Population 49

6. Favorite U.S. President for 654 students at Capitol Elementary School

Washington Lincoln Kennedy Reagan Obama TOTALSample students 7 10 4 3 26 50

7. Favorite amusement park among 1,024 students at Park Middle School

Study the information in each table below. Make at least two inferences and explain your thinking.

8. Student lunch preferences of 1,024 students at Eagle Middle School

Salad Hot Dogs Pizza TOTALSample 1 9 21 60 90Sample 2 9 18 63 90

9. Favorite fast food restaurants among 988 students at King Middle School

Burger Time Taco Place Carla’s Sandwiches R Us TOTAL

7th grade boys 28 24 12 16 807th grade girls 15 12 10 13 50

8th graders 25 18 18 39 100

10. Favorite football teams of 1,123 students at Cedar Middle School

Wolves Crows Stars Bobcats TOTAL7th grade boys 12 20 40 48 120

7th grade girls 3 6 21 45 75

8th grade boys 25 5 28 52 110

8th grade girls 21 6 54 19 100

Play Land 15 Fun Time

18

Roller World9

Water World3

50 Lesson 2-K ~ Measures Of Center And Variability In Two Data Sets

MEASURES OF CENTER AND VARIABILITY IN TWO DATA SETS

LESSON 2-K

It is helpful to have one number that describes a data set when comparing it with another data set. Which number should be used?

You might choose to compare two data sets using measures of center. Measures of center include mean, median and mode. The mean is the average of the numbers. The median is the number in the middle of the ordered data set. The mode is the number or numbers that occur most often. Depending on the numbers in the data set, there can be one mode, multiple modes or no mode.

Carl and Anna wanted to compare their test scores. Carl has taken 9 tests and Anna has taken 10 tests. Compare the means, medians and modes of their two sets of test scores.

Carl 85, 81, 93, 60, 75, 86, 95, 87, 85

Anna 78, 96, 96, 84, 84, 73, 98, 100, 76, 85

Find each mean. Carl: 85+81+93+60+75+86+95+87+85 _______________________ 9 = 747 ___ 9 = 83

Anna: 78+96+96+84+84+73+98+100+76+85 ___________________________ 10 = 870 ___ 10 = 87

The mean of Carl’s test scores is 83. The mean of Anna’s test scores is 87. They differ by 4 points.

EXAMPLE 1

solution

Lesson 2-K ~ Measures Of Center And Variability In Two Data Sets 51

Find each median. Carl: 60, 75, 81, 85, 85, 86, 87, 93, 95Order the numbers from → → → → ← ← ← ←least to greatest. Identifythe middle number(s). Anna: 73, 76, 78, 84, 84, 85, 96, 96, 98, 100 → → → → ← ← ← ←

84 and 85 are the middle 84 85 84.52+

=numbers. Find the mean of these numbers.

The median of Carl’s test scores is 85. The median of Anna’s test scores is 84.5. They differ by a score of 0.5.

Find each mode. Carl: 85 appears twice Anna: 84 and 96 both appear twice

The mode of Carl’s test scores is 85. The modes of Anna’s test scores are 84 and 96. Carl’s mode is similar to his mean and median. Anna has multiple modes and 96 is not similar to her mean or median. The mode is not often the best number to use to summarize data.

Carl’s test score of 60 is not typical of his other scores. It is much lower than his other scores and spreads his data out further from the mean. Because of this, the median is a better number to use than the mean as a single number to describe his scores. It tells you that 50% of his scores were above the median score and 50% were below.

At times the measures of center do not best compare two data sets. You might compare two data sets using measures of spread such as range, interquartile range or mean absolute deviation. These are also called measures of variability. Measures of variability help determine how spread apart the numbers in a data set are and give additional meaning to the measures of center (mean, median and mode).

It is helpful to first find the five-number summary for a set of data before finding its measures of variability.

The word “quartile” refers to how the data is separated into quarters. Twenty-five percent of the data falls between each pair of values.

Minimum MaximumMedianQ1 Q3

25% 25%25% 25%

EXAMPLE 1(CONTINUED)


Carl wanted to find the five-number summary for his test scores given in Example 1. Find the five-number summary of the test scores following the steps below. Carl: 85, 81, 93, 60, 75, 86, 95, 87, 85

1. Put the numbers in order and find the median.

60, 75, 81, 85, 85, 86, 87, 93, 95median

2. Find the median of the lower half of the data (this is called the 1st Quartile). If there are two numbers in the middle, include one in each half of the data.

60, 75,|81, 85, 85, 86, 87, 93, 95 78 Q1

3. Find the median of the upper half of the data (this is called the 3rd Quartile).

60, 75,|81, 85, 85, 86, 87,|93, 95 90 Q3

4. Identify the minimum and maximum values.

60, 75,|81, 85, 85, 86, 87,|93, 95 min max

The five-number summary of Carl’s test scores is 60 ~ 78 ~ 85 ~ 90 ~ 95. This gives a picture of the spread of his data.

Find the five-number summary for Anna’s test scores: 78, 96, 96, 84, 84, 73, 98, 100, 76, 85

Put the numbers in order and 73, 76, 78, 84, 84,|85, 96, 96, 98, 100find the median. 84.5 median

Find the 1st Quartile (Q1). 73, 76, 78, 84, 84,|85, 96, 96, 98, 100 If there are two numbers Q1in the middle, include one in each half of the data.

Find the 3rd Quartile (Q3). 73, 76, 78, 84, 84,|85, 96, 96, 98, 100 Q3

Find the minimum and maximum. 73, 76, 78, 84, 84,|85, 96, 96, 98, 100 min max

The five number summary is 73 ~ 78 ~ 84.5 ~ 96 ~ 100.

EXAMPLE 2

solution


The information already found to compare Carl and Anna’s test scores is summarized below.

Carl AnnaTest Scores (in order)

60, 75, 81, 85, 85, 86, 87, 93, 95 73, 76, 78, 84, 84, 85, 96, 96, 98, 100

Mean 83 87

Median 85 84.5

Mode(s) 85 84 and 96

Five-Number Summary

60 ~ 78 ~ 85 ~ 90 ~ 95 73 ~ 78 ~ 84.5 ~ 96 ~ 100

Measures of variability show other ways to compare the test scores. The range, interquartile range and mean absolute deviation for Carl’s and Anna’s test scores can be found using the steps below.

Step 1: Copy the chart below.

Carl Anna

Range

Interquartile Range

Mean Absolute Deviation

Step 2: Carl and Anna want to know the range of their data sets. The range is the difference between the maximum value in the data set and the minimum value in the data set. a. Find the minimum value and the maximum value in each data set. b. Subtract the minimum value from the maximum value for Carl and again for Anna. These values represent the range of each data set. Write the values in the chart.

Step 3: Another way to see spread in a data set is to look at its interquartile range. The interquartile range is the range of the middle half of the data. It excludes outliers in the data. a. Find the value for the 1st Quartile (Q1) and the 3rd Quartile (Q3) in the five-number summaries for Carl and Anna. b. Find the difference between the 3rd Quartile (Q3) and the 1st Quartile (Q1) for Carl and again for Anna. These are the interquartile ranges for each data set. Write these values in the chart.

Step 4: Another way to measure the spread of data is to see how far each number is from the mean of the data set. The average of these differences is the mean absolute deviation. a. A chart helps organize the work when finding the mean absolute deviation. There is a chart on the next page for Carl and Anna. Carl’s chart is filled in. Copy and complete the chart for Anna’s data.

EXPLORE! COMPARING TESTS


Carl

Test Score Deviation from Mean

Absolute Deviation

60 60 − 83 = −23 2375 75 − 83 = −8 881 81 − 83 = −2 285 85 − 83 = 2 285 85 − 83 = 2 286 86 − 83 = 3 387 87 − 83 = 4 493 93 − 83 = 10 1095 95 − 83 = 12 12

Anna

Test Score Deviation from Mean

Absolute Deviation

737678848485969698

100

b. Find the sum of the absolute deviations for Carl. c. Divide the sum of the absolute deviations for Carl by the number of tests taken (9). Round your answer to the nearest hundredth. This is the mean absolute deviation for Carl. Write it in the chart from Step 1. d. Complete the chart for Anna. Repeat parts b-c to find the mean absolute deviation for Anna.

Step 5: Write at least two complete sentences comparing Carl’s and Anna’s test scores.

A small mean absolute deviation tells you there is less spread in the data set. It means most or all of the numbers are close to the mean. When the mean absolute deviation is larger, the numbers are more spread out from the mean and there is greater variability in the numbers in the data set.

EXPLORE! (CONTINUED)


Find the mean absolute deviation for the data set 5, 6, 8, 10, 11

Find the mean. 5 + 6 + 8 + 10 + 11 _____________ 5 = 40 __ 5 = 8

Organize the data in a chart to Value

Deviation from Mean

Absolute Deviation

5 5 – 8 = –3 36 6 – 8 = –2 28 8 – 8 = 0 0

10 10 – 8 = 2 211 11 – 8 = 3 3

find the deviation from the meanand the absolute deviation.

Find the mean of the absolute deviations. 3 2 0 2 3 105 5

+ + + += = 2

The mean absolute deviation is 2. There is some variability from the mean.

EXERCISES

Find the mean, median, and mode(s) for each data set. If there is no mode, state “no mode.” 1. 7, 17, 35, 19, 14, 24, 17 2. 45, 31, 55, 31, 51, 31, 55, 23, 31, 27

3. 82, 92, 78, 76, 80, 86, 90, 80 4. 9, 15, –5, 12, –7, 10, 8

Find the mean, median and mode(s) for each data set. Write a sentence that compares each measure of center between the pairs of data sets.

5. Mark’s test scores: 82, 83, 74, 94, 76, 71, 94 Irina’s test scores: 95, 83, 79, 95, 82, 79, 89

6. Number of bowling pins knocked over by each roll of the bowling ball in a single game. Manuel: 7, 3, 5, 3, 10, 10, 8, 0, 7, 1, 9, 0, 8, 0, 10, 7, 3, 8 Isabel: 5, 4, 10, 0, 7, 10, 8, 2, 6, 6, 3, 4, 3, 9, 0, 8, 0

EXAMPLE 3

solution


Find the mean, median and mode(s) of each data set. Which number best describes the values in the data set? Explain why.

7. Yearly Salaries: $200,000 $70,000 $65,000 $68,000 $60,000 $30,000 $62,000 $70,000

8. Number of points in 7 games for a basketball team: 98, 87, 85, 102, 85, 93, 94

9. Home Prices: $1,000,000 $300,500 $320,000 $290,000 $250,000

10. Shoe sizes sold for a tennis shoe: 6, 7, 7, 7.5, 8, 8.5, 8.5, 8.5, 8.5, 9

11. Carolyn found the median for a set of data. Her work is below. She made a mistake. What was her mistake? Correct her work and find the median of the data set.

10 12 8 15 20 24 → → ← ← median = 8 + 15 _____ 2 = 11.5

For Exercises 12–15, find the following measures of variability: a. five-number summary b. range of the data c. interquartile range of the data 12. 4, 6, 8, 12, 20, 40 13. 24, 20, 27, 30, 24

14. 12, 17, 12, 20, 16, 12, 8, 19, 18, 6 15. 74, 76, 79, 68, 51, 59, 76, 60, 78

Find the mean absolute deviation of each data set.

16. 2, 10, 15, 20, 28 17. 15, 20, 25, 12, 21, 7, 11, 9

18. The two data sets below have the same median and mean, but not the same values. Use measures of variability to explain the differences between the two data sets.

Data Set 1: 2, 5, 11, 15, 22 Data Set 2: 10, 11, 11, 11, 12

19. The five-number-summary and mean for two data sets is shown below. Each data set has 30 values. Which data set has the least mean absolute deviation? Explain your answer.

Data Set 1: 10 ~ 20 ~ 30 ~ 40 ~ 50 mean = 32 Data Set 2: 60 ~ 62 ~ 64 ~ 66 ~ 68 mean = 65


For Exercises 20–22, find the: a. five-number summary b. range of the data c. interquartile range of the data d. mean absolute deviation of the data e. Compare the results and write sentences to compare the two data sets.

20. Money (in dollars) earned babysitting

January February March April May June

Kelsey 40 30 15 20 20 55

Rachel 25 35 60 10 75 35

21. Number of football passes completed in regular season games

Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Week 8 Week 9

Wildcats 33 29 24 21 18 16 26 26 23

Tigers 32 31 30 16 24 27 15 33 26

22. Height (inches) of girls and boys in Ms. Pon’s seventh grade class.

48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78

× × ×××

××

× × ×

Height of girls (inches)

48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78

× ××

××

××

×

Height of boys (inches)

× × × ×

COMPOUND PROBABILITIES USING LISTS, TREE DIAGRAMS …mrvsmath.weebly.com/uploads/1/3/4/5/13456042/ccss_stage_2_booklet_student_edition.pdfLesson 2-G ~ Compound Probabilities Using

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