A roulette wheel has 38 slots, numbered 0, 00, and 1 to 36. The slots 0 and 00 are colored green, 18 of the others are red, and 18 are black. The dealer spins the wheel and at the same time rolls a small ball along the wheel in the opposite direction. The wheel is carefully balanced so that the ball is equally likely to land in any slot when the wheel slows. Gamblers can bet on various combinations of numbers and colors.
(a) If you bet on "red," you win if the ball lands in a red slot. What is the probability of winning with a bet on red in a single play of roulette?
(b) You decide to play roulette 4 times, each time betting on red. What is the distribution of X, the number of times you win?
(c) If you bet the same amount on each play and win on exactly 2 of the 4 plays, you will "break even." What is the probability that you will break even?
(d) If you win on fewer than 2 of the 4 plays, you will lose money. What is the probability that you will lose money?