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2 March, 15:42

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.

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  1. 2 March, 17:47
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    (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
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