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!)