Expected Value 3.1. On April 14, 1993, during halftime of a basketball game between the. One-and-One Free-Throws

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1 ! Expected Value On April 14, 1993, during halftime of a basketball game between the Chicago Bulls and the Miami Heat, Don Calhoun won $1 million by making a basket from the free-throw line at the opposite end of the court. Don was chosen at random to attempt the basket as part of a promotional contest. A Sports Illustrated article explained: The odds against one randomly chosen person given one shot from the opposite foul line and making it are considered astronomical. Scottie Pippen admitted that after practice one day he and Michael Jordan tried to hit the shot but couldn t. (Source: Bessone, Lisa, Sports People: Don Calhoun. Sports Illustrated, April 26, 1993, vol 48.) Fortunately, not every basket is this difficult to make! In this Investigation, you will use a player s free-throw percent to figure out what is likely to happen in a given free-throw situation. 3.1 One-and-One Free-Throws In the district finals, Nishi s basketball team is 1 point behind with 2 seconds left. Nishi has just been fouled, and she is in a one-and-one free-throw situation. This means that Nishi will try one free throw. If she makes it, she tries a second free throw. If she misses the first time, she does not get a second try. Nishi s free-throw average is 60%. 38 What Do You Expect?

2 Getting Ready for Problem 3.1 What are the possible scores Nishi can make in a one-and-one free-throw situation? How can each score be made? How would you design an experiment to analyze this situation? Problem 3.1 Simulating a Probability Situation A. Is it most likely that Nishi will score 0 points, 1 point, or 2 points? Record what you think before you analyze the situation. B. Use this spinner to simulate Nishi s one-and-one situation 20 times. Record the result of each trial. Hit Miss C. Based on your results, what is the experimental probability that Nishi will score 0 points? That she will score 1 point? That she will score 2 points? D. Make an area model for this situation using a grid. What is the theoretical probability that Nishi will score 0 points? That she will score 1 point? That she will score 2 points? Compare the three theoretical probabilities with the three experimental probabilities. E. Suppose Nishi s free-throw average is 70%. How does this affect the outcome? Explain. F. How does the diagram in Question D help you predict how many times Nishi will score 2 points in 100 one-and-one situations? In 200 one-and-one situations? Investigation 3 Expected Value 39

3 3.2 Finding Expected Value In Problem 3.1, you looked at the probabilities of different outcomes of Nishi s one-and-one free-throw situation. You might have been surprised about which outcome is most likely. In this Problem, you will look at the number of points Nishi can expect to make each time she is in a one-and-one free-throw situation. Problem 3.2 Finding Expected Value Suppose Nishi has a 60% free-throw percentage and is in a one-and-one free-throw situation 100 times during the season. A. 1. How many times can she expect to score 0 points? What is the total number of 2. How many times can she expect to score 1 point? What is the total number of 3. How many times can she expect to score 2 points? What is the total number of 4. What total number of points do you expect Nishi to score in these 100 situations at the free-throw line? 5. What would Nishi s average number of points (expected value) per situation be? 40 What Do You Expect?

4 B. 1. In a one-and-one situation, what score is most likely to happen for a player whose free-throw percentage is 20%? 40%? 60%? 80%? 2. Copy and complete the table at the right for the players in part (1) in 100 one-and-one situations. You will fill in the expected value column in part (3). 3. Calculate the average number of points for each situation. Record these values in the table in part (2). Describe any pattern you see. Player s Free-Throw Percentage 20% 40% 60% 80% Points Expected in 100 One-and-One Situations Points Expected Value, or Average C. 1. Make a graph like the one started at the right for each of the players in your table. Use your graph to answer parts (2) (4). 2. How does the average number of points compare for players with a 20% free-throw percentage, a 40% free-throw percentage, a 60% free-throw percentage, and an 80% free-throw percentage? 3. Nishi s dad noticed that he makes an average of about 1 point in one-and-one free-throw situations. What is his free-throw percentage? 4. Nishi s older sister has a 70% free-throw percentage. What is her average number of points in one-and-one situations? Check your answer by making an area model. Average Points per Situation 2 1 Points Expected in a One-and-One Situation % 40% 60% 80% Free-Throw Percentage 3.3 Choosing Pay Plans The average number of points in the one-and-one situation is the expected value for this situation. Expected value is used in a variety of other situations such as determining life-insurance premiums and lottery prizes, and for predicting gains and losses in businesses. The following problem shows another situation in which expected value is useful. Investigation 3 Expected Value 41

5 Problem 3.3 Comparing Expected Values Julie and Brandon cut lawns for their neighbors to earn money to donate to a local charity. They thought that customers should pay $20. However, several customers offered a different pay plan and asked the students to choose. Julie and Brandon have to decide which of the pay plans would give them a fair deal over the long run. In each case, calculate the expected value. Then, decide whether they should accept or reject the pay plan. Give reasons for your decision. A. The customer spins each of these spinners once. If one spinner is red and the other is blue (the order makes no difference), Julie and Brandon get $24. Otherwise, they get $10. Green Blue Green Red Red Yellow Blue B. The customer rolls a pair of number cubes and adds the two numbers. If the sum is even, Julie and Brandon get $25. If it is odd, they get $18. C. The customer rolls a pair of number cubes. If the product is 24, Julie and Brandon get $40. Otherwise, they get $10. D. The customer places a twenty-dollar bill and three one-dollar bills in a bag. Julie and Brandon choose two bills. The money they choose is their payment. E. The customer tosses three coins. If two or more land heads up, Julie and Brandon get $25. Otherwise, they get $ What Do You Expect?