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Say you have a digital slot machine. Rather than using virtual reels, this slot machine generates results using predetermined probabilities for a given symbol appearing in any position.

Unlike normal symbols, scatter symbols in any visible position count towards a match.

Given a scatter symbol's probability of appearing in any given position and that there are fifteen positions where a symbol can appear (five 'reels'/columns and three visible rows), how does one calculate the probability of each possible number of symbols (0-15) appearing?

Please include in your response a formula which is readable by a layman. (I'm no mathemetician!)

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I've found the solution, thanks to this article on Binomial Probability. It is necessary in this case to think of the slot machine not as 5 reels with 3 positions visible each, but as 15 reels with 1 position visible each.

Given:

  • N: The number of symbols we are looking for
  • P: The probability of the symbol appearing in any given position
  • R: The number of reels

The odds of N symbols appearing is:

$\binom{R}{N} * P^N * (1-P)^{R - N}$

So for 15 reels:

$\binom{R}{N} * P^N * (1-P)^{15 - N}$

e.g. 2 Jackpot symbols when this symbol appears 1% of the time:

$\binom{15}{2} * 0.01^2 * 0.99^{15 - 2} = $ ~0.9214%

For the layman out there (like myself), $\binom{R}{N}$ represents a combination. I found a nice explanation of combinations here.

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