Chapter 7.2, Problem 18PB

Playing roulette A roulette wheel in Las Vegas has 38 slots. If you bet a dollar on a particular number, you’ll win $35 if the ball ends up in that slot and $0 otherwise. Roulette wheels are calibrated so that each outcome is equally likely.

1. a. Let X denote your winnings when you play once. State the probability distribution of X. (This also represents the population distribution you would get if you could play roulette an infinite number of times.) It has mean 0.921 and standard deviation 5.603.
2. b. You decide to play once a minute for 12 hours a day for the next week, a total of 5040 times. Show that the sampling distribution of your sample mean winnings has mean = 0.921 and standard deviation = 0.079.
3. c. Refer to part b. Using the central limit theorem, find the probability that with this amount of roulette playing, your mean winnings is at least $1, so that you have not lost money after this week of playing. (Hint: Find the probability that a normal random variable with mean 0.921 and standard deviation 0.079 exceeds 1.0.)