131(8) A standard slot machine has three reels. The first reel has 4 walnuts, 7

Question

131(8) A standard slot machine has three reels. The first reel has 4 walnuts, 7 cherries, 3 oranges, 4 lemons. I bell, and 1 jackpot bar. The second reel has 7 cheries, 6 oranges, 1 lemon, 3 bells, and 3 bars. The thied reel has 4 walnuts, 7 oranges, 5 lemons, 3 bells and 1 bar. The payoffs are: Reel 1 Payoft in Coins 85 25 20 14 Reel 2 Bar Bar Bar Bell Bell Bar Bell Bell Bell Lemon Lemon Bell Lemon Lemon Lemon Orange Orange Bar Orange Orange Orange Cherry Cherry Bell Cherry Cherry Walnut Reel 3 10 10 Cherry Cherry Any other (a) What is the expected payoff 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 slot machine) keep?

Explanation / Answer

a. The table below lists the possible scenarios for payoff and the probability of occuring of them

b. So, in the long run, expected payoff from game = 0.031875 + 0.009375 + .. + 0.196 = 0.81525

So, the machine keeps 1-0.81525 = 0.18475 dollars for every dollar spent on it.

Reel1 Reel2 Reel3 Probability Payoff Expected payoff Bar Bar Bar 0.05*0.15*0.05 85 0.031875 Bell Bell Bar 0.05*0.15*0.05 25 0.009375 Bell Bell Bell 0.05*0.15*0.15 20 0.0225 Lemon Lemon Bell 0.2*0.05*0.15 14 0.021 Lemon Lemon Lemon 0.2*0.05*0.25 16 0.04 Orange Orange Bar 0.15*0.3*0.35 10 0.1575 Orange Orange Orange 0.15*0.3*0.15 10 0.0675 Cherry Cherry Bell 0.35*0.35*0.15 8 0.147 Cherry Cherry Walnut 0.35*0.35*0.2 5 0.1225 Cherry Cherry Any 0.35*0.35*0.8 2 0.196

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