EXAMPLE 25 Lou Gets Entertainment

Return to Example 22 (page 378) and to Lou, who bets on red at the roulette wheel. Figure 8.25 shows the probability model and corresponding probability histogram for Lou’s favorite bet.

The probability model in Figure 8.25 is discrete, with just two possible outcomes: win $1 or lose $1. Yet the central limit theorem says that the average outcome of many bets follows a normal curve. Lou is a habitual gambler who places 50 bets of $1 on red almost every night. Because we know the probability model for a bet on red, we can simulate Lou’s experience over many nights at the roulette wheel. The histogram in Figure 8.26, made from a simulation of 1000 nights, shows Lou’s average winnings per bet, , from bets.

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As the central limit theorem says, the distribution looks normal and all we need to completely specify the normal curve shown in Figure 8.26 is its mean and standard deviation. For that we return to the distribution of outcomes for one bet on red in Figure 8.25. From Example 22 (page 378) and Self Check 16 (page 380) we know that the mean and standard deviation are , respectively. Next, we use the information from the central limit theorem to specify the mean and standard deviation of the average outcome, , from bets.

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Figure 8.25: Figure 8.25 The probability model and probability histogram for betting on red at the roulette wheel.
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Figure 8.26: Figure 8.26 (Approximately normal) histogram of a gambler’s winnings from a simulation of 1000 nights, where each night had 50 bets on red (or black) in roulette.

Now that we have completely specified the approximate distribution of , we can apply the 99.7 part of the 68—95—99.7 rule from Section 5.9 (page 216): Almost all average nightly winnings per bet will fall within 3 standard deviations of the mean, that is, between

and

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This gives us a 99.7% confidence interval for Lou’s mean winnings per bet, : between .

What is more interesting to Lou is not average winnings per bet, but total winnings for the whole 50-bet night. To find this out, Lou can simply multiply both endpoints from the preceding 99.7% confidence interval by 50. So Lou’s total winnings after 50 bets of $1 each will almost surely fall between

and

Each night, Lou may win as much as $18.50 or lose as much as $23.80. Note that he will usually lose more on a bad night than he will win on a good night. Some people find gambling exciting because the outcome, even after an evening of bets, is uncertain. It is possible to beat the odds and walk away a winner. It’s all a matter of luck.