A tourist in Las Vegas was attracted by a certain gambling game in which the customer stakes 1 dollar on each play; a win then pays the customer 2 dollars plus the return of her stake, although a loss costs her only her stake. Las Vegas insiders, and alert students of probability theory, know that the probability of winning at this game is 1/4. When driven from the tables by hunger, the tourist had played this game 240 times. Assuming that no near miracles happened, about how much poorer was the tourist upon leaving the casino? What is the probability that she lost no money?

We have seen that, in playing roulette at Monte Carlo (Example), betting 1 dollar on red or 1 dollar on 17 amounts to choosing between the distributions

or

You plan to choose one of these methods and use it to make 100 1-dollar bets using the method chosen. Using the Central Limit Theorem, estimate the probability of winning any money for each of the two games. Compare your estimates with the actual probabilities, which can be shown, from exact calculations, to equal .437 and .509 to three decimal places.

In Example find the largest value of [math]p[/math] that gives probability .954 that the first decimal place is correct.

It has been suggested that Example is unrealistic, in the sense that the probabilities of errors are too low. Make up your own (reasonable) estimate for the distribution [math]m(x)[/math], and determine the probability that a student's grade point average is accurate to within .05. Also determine the probability that it is accurate to within .5.

Find a sequence of uniformly bounded discrete independent random variables [math]\{X_n\}[/math] such that the variance of their sum does not tend to [math]\infty[/math] as [math]n \rightarrow \infty[/math], and such that their sum is not asymptotically normally distributed.