Question

Does anyone know to create a model of a ball and urn model of an American Roulette Wheel? Please provide a detailed model with an explanation of how you came to the conclusion you did. And if possible, provide an example of using the model to solve a problem. Thank you!

Answer 1

American Roulette Wheel is a casino game. In each play of roulette, a croupier spins a roulette wheel (consists of coloured pockets numbered 1 through 36, 0 and a double 0) in one direction, then rolls a ball in the opposite direction and the ball falls into one of the pockets.

Players can bet on whether the ball will fall into the pockets numbered 1–36, but not on a pocket that belongs to the house (0 and 00). The croupier pays the winnings to those player(s) who successfully wins the bet.

One needs to remember that each spin(or trial) is independent of each other.

Suppose a player bets on the numbers 1 through 18. Then the probability of winning the bet is p=18/38. And this probability of winning or success remains the same for any number of spins or trials. And since each trial is independent, the probability that the player will win or lose 'x' times in the first 'k' trials can be obtained using a binomial model.

The probability that the player will win twice in the first 5 spins can be calculated as below:-

n=5, p=0.47

P(X=x) = [nCx]* [(p)^x]* [(1-p)^(n-x)]

P(X=2) = [5C2]* [(0.47)^2]* [(1-0.47)^(5-2)]

P(X=2) = 0.33

Hence, the probability that the player will win twice in the first 5 spins is 0.33.

Similarly the x^th win at the k^th spin can be modeled using Negative binomial distribution. If x =1 (i.e., the first win at the k^th spin) can be modeled using the geometric distribution.