# How to calculate the payout rate of a slot machine?

I am interested in learning how to calculate the payout rate of a slot-machine.

Imagine a machine with $s$ different symbols on $r$ reels. Each symbol has a given payout, $p$, if the symbols end up next to each other.

I know basic probability- number of chances by number of possibilities. But, I don't know how to get to the number of chances and possibilities.

Thanks!

• What do you mean by "the symbols end up next to each other"? You mean that each of the $r$ reels ends up with the same symbol (out of the $s$) showing? Jun 28, 2011 at 15:26
• exactly. like the slot machines you maybe know from casinos. they win a certain price relative to your bet if you have three 7s on the middle row. Jun 28, 2011 at 15:27

You identify each payout and the probability of it, and add them up. So if there is one jackpot, requiring you get a specific symbol on all the reels, that pays $1,000,000$ (in units of the bet), the chance you get it is $1$ in $s^r$. This win contributes $\frac{1,000,000}{s^r}$ to the payout rate. If you have another one that requires a single symbol on any reel that pays 3, it contributes $3\left({1-\left(\frac{s-1}{s}\right)^r}\right)$. You just have to count the win chances, multiply by the payout, and add them up. One challenge is making sure they are mutually exclusive-in my example if the jackpot symbol and the one in the win paying 3, you would have to subtract the jackpot probability from the other, as you presumably only pay the jackpot.