On the popular television game show The Price is Right, contestants can play “The Showcase Showdown.” The game involves a large wheel with 20 nickel values, 5, 10, 15, 20, ..., 95, 100, marked on it. Contestants spin the wheel once or twice, with the objective of obtaining the highest total score without going over a dollar (100). [According to the American Statistician (Aug. 1995), the optimal strategy for the first spinner in a three-player game is to spin a second time only if the value of the initial spin is 65 or less.] Let x represent the score of a single contestant playing “The Showcase Showdown.” Assume a “fair” wheel (i.e., a wheel with equally likely outcomes). If the total of the player’s spins exceeds 100, the total score is set to 0.

a. If the player is permitted only one spin of the wheel, find the probability distribution for x.

b. Refer to part a. Find E(x) and interpret this value.

c. Refer to part a. Give a range of values within which x is likely to fall.

d. Suppose the player will spin the wheel twice, no matter what the outcome of the first spin. Find the probability distribution for x.

e. What assumption did you make to obtain the probability distribution in part d? Is it a reasonable assumption?

f. Find m and s for the probability distribution of part d, and interpret the results.

g. Refer to part d. What is the probability that in two spins the player’s total score exceeds a dollar (i.e., is set to 0)?

h. Suppose the player obtains a 20 on the first spin and decides to spin again. Find the probability distribution for x.

i. Refer to part h. What is the probability that the player’s total score exceeds a dollar?

j. Given that the player obtains a 65 on the first spin and decides to spin again, find the probability that the player’s total score exceeds a dollar.

k. Repeat part j for different first-spin outcomes. Use this information to suggest a strategy for the one-player game.