Suppose you play roulette a number of times, each time betting$1 on red (recall

Question

Suppose you play roulette a number of times, each time betting$1 on red (recall that there are 18 slots on a roulette wheel, anda bet on red pays 1 to 1). a.) What is the probability of making a net gain in 100plays? b.) What is the probability of making a net gain in 1,000plays? Suppose you play roulette a number of times, each time betting$1 on red (recall that there are 18 slots on a roulette wheel, anda bet on red pays 1 to 1). a.) What is the probability of making a net gain in 100plays? b.) What is the probability of making a net gain in 1,000plays?

Explanation / Answer

An American roulette wheel has 38 numbers, 18 of which arered. Thus, p(red) = 18/38 = 9 / 19. Let's see if the Central Limit Theorem applies: n*p = 100 * 9/19 = 47.37 > 5 n*(1-p) = 100 * 10/19 = 52.63 > 5 Thus, we can use it. Let x = number of red successes. x ~ N(np, sqrt(n*p*(1-p)) =N(47.37, 4.993) We want to find the probability of making a gain on 100 plays (i.e.x > 50) P(X > 50) =1 - P(X < 50) z = (50 - 47.37) / 4.993 = .527 --> p = .7019 1 - .7019 = .2981 Thus, the probability of making a net gain in 100 plays is.2981. With 1000 trials, x ~ N(np, sqrt(n*p*(1-p))) = N(473.68,15.789) P(X > 500) = 1 - P(X p = .9525 1-.9525 = .0475. Thus, the probability of making a net gain in 1000 plays is.0475.

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