Deciding to put probability theory to good use, we encounter a slot machine with

Question

Deciding to put probability theory to good use, we encounter a slot machine with three independent wheels, each producing one of the five symbols BAR, BELL, ORANGE, LEMON, or CHERRY with equal probability. The slot machine has the following payout scheme for a bet of 1 coin (where “?” denotes that we don’t care what comes up for that wheel): BAR/BAR/BAR pays 25 coins

BELL/BELL/BELL pays 10 coins

ORANGE/ORANGE/ORANGE pays 5 coins

LEMON/LEMON/LEMON pays 4 coins

CHERRY/CHERRY/CHERRY pays 3 coins

CHERRY/CHERRY/? pays 2 coins

CHERRY/?/? pays 1 coin

1.Compute the expected “payback” percentage of the machine. In other words, for each coin played, what is the expected coin return?

2. How many coins can the casino offer as the “jackpot” (BAR/BAR/BAR) without (statistically) losing money?

3. Compute the probability that playing the slot machine once will result in a win (defined as winning anything, including breaking even).

4. Estimate the mean and median number of plays you can expect to make until you go broke, if you start with 10 coins. You can run a simulation to estimate this, rather than trying to compute an exact number.

Explanation / Answer

Solution

Back-up Theory

1. ‘three independent wheels’ => joint probability = product of individual probabilities. 2. ‘each producing one of the five symbols BAR, BELL, ORANGE, LEMON, or CHERRY with equal probability.’ => P(BAR) = P(BELL) = …… = P(CHERRY) = 1/5

3. Expected payback = coin return x probability of coin return, where coin return = Number of payout coins – 1(the one inserted to play)

Now, to work out the solution,

Possibility                                 Payout    Payback    Probability    

                                                 (coins)     (coins)   

BAR/BAR/BAR                            25            24           1/125

BELL/BELL/BELL                        10              9          1/125             

ORANGE/ORANGE/ORANGE      5             4          1/125

LEMON/LEMON/LEMON              4               3           1/125

CHERRY/CHERRY/CHERRY       3               2          1/125   

CHERRY/CHERRY/?                    2               1            1/25

CHERRY/?/?                                 1               0             1/5

Expected Payback = (1/125)(42) + (1/25)(1) + (1/5)(0) = 47/125 or 37.6% ANSWER

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