31(8) A standard slot machine has three reels. The first reel has 3 walnuts, 7 c

Question

31(8) A standard slot machine has three reels. The first reel has 3 walnuts, 7 cherries, 3 oranges, 5 lemons bell, and 1 jackpot bar. The second reel has 7 cherries, 6 oranges, 1 lemon, 3 bells, and 3 bars. The third eel has 4 walnuts, 7 oranges, 5 lemons, 3 bells and 1 bar. The payoffs are: Payoff in Coins 85 18 18 14 14 10 10 Reel 1 Reel 2 Reel 3 Bar Bar Bar Bell BellBar Bell Bell Bell Lemon Lemon Bell Lemon Lemon Lemon Orange Orange Bar Orange Orange Orange Cherry Cherry Bel Cherry Cherry Walnut Cherry Cherry Any other (a) What is the expected payof if each picture on each reel is equally likely to appear? (b) In the long run, what fraction of every dollar bet will this one-armed bandit (the slop machine) keep?

Explanation / Answer

You have to find the probabilities of each combination. Then multiply each by its payoff. Finally add them all up.

Bar-bar-bar = (1/20)(3/20)(1/20) = 3/8000 * 85 = 0.031875
Bell-bell-bar = (1/20)(3/20)(1/20) = 3/8000 * 18 = 0.00675
Bell-bell-bell = (1/20)(3/20)(3/20) = 9/8000 * 18 = 0.02025
Lemon-Lemon-Bar = (5)(1)(1)/8000 = 5/8000 * 14 = 0.00875
Lemon-Lemon-Lemon = (5)(1)(5)/8000 = 25/8000 * 14 = 0.04375
Orange-Orange-bar = (3)(6)(1)/8000 = 18/8000 * 10 = 0.0225
Orange-Orange-Orange = (3)(6)(7)/8000 = 126/8000 * 10 = 0.1575
Cherry-Cherry-Bell = (7)(7)(3)/8000 = 147/8000 * 5 = 0.091875
Cherry-Cherry-Walnut = (7)(7)(4)/8000 = 196/8000 * 5 = 0.1225
Cherry-Cherry-Any = (7)(7)(20)/8000 = 980/8000 * 3 = 0.3675

a ) Adding them up gives an expected value of 0.87325. This is the amount you can expect to win for every dollar bet .

b) The machine wins 1-0.87325 = 0.12675 or 12.675 % of every dollar bet.

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