Roulette is a game named after a French diminutive for little wheel. In the game, players may choose to place bets on either a single number or a range of numbers, the colors red or black, or whether the number is odd or even. The 18 red, 18 block, and 2 green pockets of the roulette wheel are numbered from 1 to 36, 0, and 00. The payout is 1 to 1 if you bet on red or block, 1 to 1 if you bet on even or odd numbers, 35 to 1 if you be on one number among 38 available numbers.
(a) Suppose the outcome of interest is the colour of the pocket in which the ball falls. List all possible outcomes for three games. Are the elements in the sample space equally likely?
(b) Find the probability distribution of the number of games that the ball falls into a red pocket for three games.
(c) Let X be the number of games that the ball falls into a even-numbered pocket in the next 20 games. Verify that X is a binomial random variable. Note that 0 and 00 don’t count as an even number in this game.
(d) Determine the mean and variance of X.
(e) What is the probability that there will be no more than 17 games that the ball falls into a red pocket in the following 20 games.