Defining casino pay tables we can understand

I may be giving away one of the greatest trade secrets I know. I hope this doesn’t come back to haunt me.
I realize I have a strong math background, but when I started in this industry, I didn’t realize what I’m about to show you is not universally understood. It is how to calculate the payback of a pay table.
When an inventor asks me to provide frequencies of hands for side bets, I quickly provide them. In return, I get a document that asks me to verify their payback calculation. This, inevitably leads me to ask them – What did you do???
It simply isn’t that hard. You take the frequency (i.e. probability) of a particular hand and multiply it by the payout of that hand. This gives us the contribution rate of that hand. If the hand pays 5 to 1, we use 6 because the payout includes the return of the original wager. When we add up the contribution rates of all the winning hands, we get the payback.
No need to use squares, square roots, imaginary numbers or pi. It is a very simple formula. To illustrate, let’s use a common pay table for Pair Plus:
To get the frequency of each hand, I used a program to determine the number of occurrences of each hand (they can also be calculated using combinatorial math). I divide these occurrences by 22,100, which is the number of unique hands of three cards from a 52-card deck to get the frequencies.
I then multiply each of these frequencies by the payout. As the payouts for Pair Plus are “to 1,” the numbers above represent the full-pay version of the paytable of 40, 30, 6, 4, 1. When we sum up the values, we find the hit frequency is 25.6109% (this is the sum of the frequencies of the winning hands) and the payback is 97.6833%, which is the sum of the Contribution rates.
Payback Calculation – Pair Plus
Straight Flush
Three of a Kind

* Includes the return of the original wager
The information in this simple table is very important in understanding what to expect. First of all, we can expect to win roughly 1 in 4 hands. About 2/3 of these wins will be a Pair. We should get a Flush about 1 in 20 hands and a Straight about 1 in 33. Three of a Kinds and Straight Flushes are considerably rarer, each being in the 1 in 400-500 range.
While Pairs will make up about two in three winning hands, it will account for only about 1/3 of our money won. Flushes and Straights together add up to roughly half of our expected win in monetary terms. So, if you don’t get your fair share of these, getting a couple of extra Pairs may not help your bankroll much.
Get on a hot streak of Straights and Flushes and you could be heading for a good night.
What is also quickly learned from this table is the impact of reducing the payout of any particular hand. If the Flush only pays 3 instead of 4, then we will multiply its frequency by 4 instead of 5, resulting in a decrease of the contribution rate for Flush, and in turn the overall payback by almost 5%.
If we decrease the Straight payout by 1 it will reduce the payback by 3.25%. If a casino chooses to pay out 50 for the Straight Flush, this will increase the payback by about 2.17% (10 times the frequency of 0.2172%).
If you remember the rough probability of each of these hands, you can quickly compute the approximate payback of the Pair Plus pay table that you are playing at. This process is not unique to Pair Plus. It applies to virtually any game with a pay table.
In the case of head to head games, against the dealer, the computation might get more complex because you have to take into account the frequency of beating the dealer and the frequency of a final hand. It is also very applicable to video poker and can be used to help a Player quickly determine if a particular pay table is worth playing.
In the case of Video Poker there is also additional complexity. Because the Player’s strategy effects the distribution of final hands, the actual frequency is a bit of a moving target. For the sake of “ballparking” the impact of a pay table change, using some baseline numbers works quite well.
I’ll cover this topic more fully in an upcoming column.

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