Question

The roulette wheel has 38 slots. Two of the slots are green, 18 are red, and 18 are black. A ball lands at random in one of the slots. A casino offers the following game. Pay $1 to enter the game. If the ball falls on black, you don’t get anything, if the ball falls on green, you get a dollar, if the ball falls on red, you get $1.95. Bob plays this game 100 times, and of course, the 100 outcomes are independent. What is the probability that he comes out ahead?

Answer 1

For each round, we compute the payoffs net of cost as:
P(X = -1) = P(black) = 18/38 = 9/19
P(X = 0) = P(green) = 2/38 = 1/19
P(X = 0.95) = P(red) = 18/38 = 9/19

Therefore the expected value and second moment of payoffs here are computed as:

E(X) = -1*(9/19) + 0.95*(9/19) = -0.0237
E(X2) = 1*(9/19) + 0.952*(9/19) = 0.9012

Therefore the standard deviation now is computed here as:

Therefore now for 100 trials, the mean of the payoffs could be modelled here as: (Using Central limit theorem)

The probability here is computed as:

Converting it to a standard normal variable, we get here:

Getting it from the standard normal tables, we get here:

Therefore 0.4014 is the required probability here.