Question: Using what you learned in this chapter, answer the following questions.
Q1. An American roulette wheel has 38 slots, of which 18 are black, 18 are red, and 2 are green. When the wheel is spun, the ball is equally likely to come to rest in any of the slots. A bet of $1 on red will win $2 if the ball lands in a red slot. (When gamblers bet on red or black, the two green slots belong to the house.) Give a probability model for the winnings of a $1 bet on red and find the expected value of this bet.
Q2. The Web site for the Mega Millions lottery game gives the following table for the various prizes and the probability of winning:
(a) The jackpot always starts at $12,000,000 for the cash payout for a $1 ticket and grows each time there is no winner. What is the expected value of a $1 bet when the jackpot is $12,000,000? If there are multiple winners, they share the jackpot, but for purposes of this problem ignore this.
(b) The record jackpot was $390,000,000 on March 6, 2007. For a jackpot of this size, what is the expected value of a $1 bet, assuming the jackpot is not shared?
(c) For what size jackpot are the expected winnings of the Mega Millions the same size as the expected winnings for roulette that you calculated in Question 1?
Q3. Do you think roulette or the Mega Millions is the better game to play? Discuss. You may want to consider the fact that, as the jackpot grows, ticket sales increase. Thus, the chance that the jackpot is shared increases. The expected values in Question 2 overestimate the actual expected winnings.