Ask the Slot Expert: Calculating a slot machine's volatility

Question: I am a junior student of math from China. I have a term project to do statistical computation of a slot game when a paytable and reel strip are given. I learned some computations from books and your website. But I got stuck when tried to compute the volatility of a slot game. I appreciate it if you could answer my questions.

Assuming the RTP of the game is R, if the game is a line game. All possible pays P_i has hit H_i, so the volatility given by a book is

VI = CI*sqrt( sum(H_i x (P_i - R)^2 ) / N)

where CI is the index of confidence level, N is the total cycle of the game, P_i could be zero (for non-winning case).

My first question is what happens if the minimum bet is not 1 credit. We know that RTP is the expect pay assuming that the bet is 1 unit credit. To make the unit of above formula consistent, we expect P_i has the same unit of R so it should be pay for 1 unit credit bet. If the minimum bet is not 1 unit, I think we should normalize the pay P_i by BET so the formula should be

VI = CI*sqrt( sum(H_i x (P_i/BET - R)^2 ) / N)

But in the book, they have

VI = CI*sqrt( sum( (H_i/BET) x (P_i - R)^2 ) / N)

I don't understand why and which one is correct.

Second, in the above formula, we consider each single pay. But in real life, each spin may have more than one winning line at the same time. So should be it more realistic to compute how many hits for each unique pay in order to calculate the VI? For example, say 5xA pays 300, 3xSCATTER pays 30, 4xQ pays 80. If all those wins occur in one screen, the total unique pay will be 300+30+80=410. We could count how many screens will give 410 pays. Then, P_i should be the total unique pay for each screen, H_i is how many screens could give that unique pay.

I think this is correct but if I computate it that way, the result is very different from that given by the first formula.

I have been thinking about it for long time but I still don't have any clue.

Answer: I've always done the the volatility calculation from the casino's perspective, not the player's. All of my casino math books also use the casino's perspective in their calculations.

The equation in my books for the value that you take the square root of is:

SUM( (NetPay_i - HE)^2 x P_i)

Where HE is the house edge at the number of coins bet and NetPay_i is the net pay per coin for payoff i at the number of coins bet.

Let's say we're playing three coins per spin. For a losing spin, the net pay per coin is 1 from the casino's perspective, (3 coins bet - 0 coins paid) divided by 3 coins played.

If the player wins 3 coins, the calculation is (3 coins bet - 3 coins paid) divided by 3 coins played for a net pay of 0.

If the player wins 300 coins, the calculation is (3 coins bet - 300 coins paid) divided by 3 coins played for a net pay of -99.

Note that Kilby and Fox in Casino Operations Management calculate the net pay by dividing the payout by the number of coins played and then subtracting 1. You then have to change the sign to get the value from the casino's perspective.

You're right that you have to take into account the number of coins bet. It has to be taken into account in the net pay figure, not in the probability.

I would like to know which book contains the formula you cited. Perhaps I'm missing something that makes their formula correct.

Moving on to your second question, I think you have the right methodology. Look at how much is won on every possible screen layout. That said, I've never seen a PAR sheet that went to that trouble. When calculating hit frequencies, the PAR sheets just multiply the single-line hit frequency by the number of coins played.

I think there are two reasons for allowing this imprecision in the calculations. First, the confidence intervals are used to give the casino a rough idea of how many plays a machine needs to have for the casino to be reasonably sure that it has made a profit on the machine. A rough idea is good enough and there will always be some uncertainty because the results are chosen at random.

And second, the calculations for CI (Student's Z score) assume a normal distribution and that's not the case for a small number of plays. Extreme accuracy just isn't needed for how the results of these calculations are used. I've never seen a PAR sheet or textbook example go to the lengths necessary to handle a multi-line machine correctly.