STATISTICS IN SUMMARY
Chapter Specifics
• The expected value is found as an average of all the possible outcomes, each weighted by its probability.
• When the outcomes are numbers, as in games of chance, we are often interested in the long-run average outcome. The law of large numbers says that the mean outcome in many repetitions eventually gets close to the expected value.
• If you don’t know the outcome probabilities, you can estimate the expected value (along with the probabilities) by simulation.
In Chapter 17, we discussed the law of averages, both incorrect and correct interpretations. The correct interpretation is sometimes referred to as the law of large numbers. In this chapter, we formally state the law of averages and its relation to the expected value. Understanding the law of large numbers and expected values is helpful in understanding the behavior of games of chance, including state lotteries. Expected values provide a way you can compare games of chance with huge jackpots but small chances of winning with games with more modest jackpots but more reasonable chances of winning.
CASE STUDY EVALUATED Using what you learned in this chapter, answer the following questions.
1. 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 (and you will also get back the $1 you bet) 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.
474
2. The website for the Mega Millions lottery game gives the following table for the various prizes and the probability of winning:
Prize: Jackpot $250,000 $10,000 $150 $150 Probability: 1 in 175,711,536 1 in 3,904,701 1 in 689,065 1 in 15,313 1 in 13,781 Prize: $10 $7 $3 $2 Probability: 1 in 844 1 in 306 1 in 141 1 in 75 (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?
3. 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.
Online Resources
• The StatBoards video Expected Value discusses the basics of computing and interpreting expected values in the context of an example.