You just got a promotion from work and decide to celebrate by spending a weekend

Question

You just got a promotion from work and decide to celebrate by spending a weekend in Vegas!
Once at the casino, you decide to test your luck at the roulette tables. A roulette wheel consist
of 18 black pockets, 18 red pockets, and 2 green pockets. You decide to just play by placing
bets on what color the ball will fall upon. You spin the wheel 8 times.
(a) What is the probability that out of the 8 spins, 3 were blacks, 3 were reds, and 2 were
greens?
(b) What is the probability that all 8 spins were greens?

Explanation / Answer

a) Each spin is either a red or not a red, each spin is either a black or not and each spin is either a green or not. Using binomial probability,

P(3 reds out of 8 spins) = (8 choose 3) * (18/38)3 + (28/38)5
P(3 blacks out of 8 spins) = (8 choose 3) * (18/38)3 + (28/38)5
P(2 green out of 8 spins) = (8 choose 2) * (2/38)2 + (36/38)6

P(3 reds, 3 blacks and 2 greens)
= [8 choose 3 *(18/38)^3 (20/38)^5] + [8 choose 3 *(18/38)^3 (20/38)^5] + [8 choose 2 *(2/38)^2 (36/38)^6]
= 0.5368

b) p(green) = 2/38, P(8 greens) = (2/38)8 = 5.888*10-11

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