A roulette wheel used in a U.S. casino has 38 slots of which 18 are red, 18 are black, and 2 are green. A roulette wheel used in a French casino has 37 slots, of which 18 are red, 18 are black, and 1 is green. A ball is rolled around the wheel and ends up in one of the slots with equal probability. Suppose that a player bets on red. If a $1 bet is placed, the player wins $1 if the ball ends up in a red slot. (The player’s $1 bet is returned.) If the ball ends up in a black or green slot, the player loses $1. Find the expected value of this game to the player in a. The United States. b. France.

(a) - $0.0526

(b) - $0.0270

Step-by-step explanation:

(a) In the United States

Total roulette slots = 38 slots, red = 18 slots, black = 18 slots, green = 2

If ball lands in the red slot, the player wins $ 1. If the ball lands in the black or green slot, the player loses $ 1.

Probability = number of events ÷ possible outcome

P (win) = P (red) = number of red slots ÷ total number of slots

P (red) = 18 ÷ 38 = 9/19

P (red) = 9/19

P (loss) = P (black or green) = number of green or black slots ÷ total number of slots

P (black or green) = P (black) + P (green) = (18 + 2) ÷ 38 = 20 ÷ 38

P (black or green) = 10/19

Expected value = sum of all (possible outcome x probability) = Value of win + Value of loss

Expected value = 1 * (9/19) + (-1) * (10/19) = (9/19) - (10/19)

Expected value = - (1/19) = - $0.0526

(b) In France

Total roulette slots = 37 slots, red = 18 slots, black = 18 slots, green = 1

If ball lands in the red slot, the player wins $ 1. If the ball lands in the black or green slot, the player loses $ 1.

Probability = number of events ÷ possible outcome

P (win) = P (red) = number of red slots ÷ total number of slots

P (red) = 18 ÷ 37 = 18/37

P (red) = 18/37

P (loss) = P (black or green) = number of green or black slots ÷ total number of slots

P (black or green) = P (black) + P (green) = (18 + 1) ÷ 37 = 19 ÷ 37

P (black or green) = 19/37

Expected value = sum of all (possible outcome x probability) = Value of win + Value of loss

Expected value = 1 * (18/37) + (-1) * (19/37) = (18/37) - (19/37)

Expected value = - (1/37) = - $0.0270