5.93 A roulette payoff revisited. Refer to the previous exercise. In part (d), the central limit theorem was used to approximate the probability that Sam ends the year ahead. The estimate was about 0.10 too large. Let’s see if we can get closer using the Normal approximation to the binomial with the continuity correction.
(a) If Sam plans to bet on 520 roulette spins, he needs to win at least $520 to break even. If each win gives him $35, what is the minimum number of wins m he must have?
(b) Given p = 1/38 = 0.026, what are the mean and standard deviation of X, the number of wins in 520 roulette spins?
(c) Use the information in the previous two parts to compute P(X ≥ m) with the continuity correction. Does your answer get closer to the exact probability 0.396?