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# How do I calculate the long-term payback of a video poker pay table?

8 February 2010

Hi, John,

I consider myself a savvy and knowledgeable video poker player, and have been playing for 11 years now; until the actual mathematics kicks in!

Can you please explain to me how we arrive at a 98.981% return (with perfect play) on a 9/6 Double Double Bonus Video Poker hand? I understand what the 9/6 means, but how do I crunch the math to show me the 98.981% return?

I would really appreciate learning the mathematical formula involved to solve and prove any return.

Thanks so much.

JG

Dear JG,

We don't really use a formula to calculate the long-term payback for a video poker paytable. We use iteration. The procedure follows.

For each 5-card hand you can be dealt (there are 2,598,960 of them), calculate the expected value of all the ways there are to draw five cards, four cards, three cards, two cards, one card and zero cards. Keep track of the option that had the highest expected value and move on the next hand. After you've evaluated all of the hands, add up the expected values to get the long-term payback for the pay table.

Let's look at how to handle a hand. Say the hand you're evaluating is a royal flush in spades. First find the value of the hand without drawing any cards. Next, find the values of all the possible hands you can make by holding the 10 of spades and drawing replacements to find the expected value of holding the 10. Then find the values of all the possible hands you can make by holding the jack of spades and drawing a replacement to find the expected value of holding the jack. Do the same for holding the queen, king and ace. Now, move on to evaluating all the possible hands you can make from all the combinations of two cards you can hold. Repeat for all of the combinations of holding three cards and four cards. The combination of hold cards with the highest expected value is the mathematically correct way to play that hand. In this case, obviously, the best play is to hold the pat royal.

Best of luck in and out of the casinos,
John

Send your slot and video poker questions to John Robison, Slot Expert, at slotexpert@comcast.net. Because of the volume of mail I receive, I regret that I can't reply to every question.

John Robison

John Robison is an expert on slot machines and how to play them. John is a slot and video poker columnist and has written for many of gaming’s leading publications. He holds a master's degree in computer science from the prestigious Stevens Institute of Technology.

You may hear John give his slot and video poker tips live on The Good Times Show, hosted by Rudi Schiffer and Mike Schiffer, which is broadcast from Memphis on KXIQ 1180AM Friday afternoon from from 2PM to 5PM Central Time. John is on the show from 4:30 to 5. You can listen to archives of the show on the web anytime.

#### Books by John Robison:

The Slot Expert's Guide to Playing Slots
John Robison
John Robison is an expert on slot machines and how to play them. John is a slot and video poker columnist and has written for many of gaming’s leading publications. He holds a master's degree in computer science from the prestigious Stevens Institute of Technology.

You may hear John give his slot and video poker tips live on The Good Times Show, hosted by Rudi Schiffer and Mike Schiffer, which is broadcast from Memphis on KXIQ 1180AM Friday afternoon from from 2PM to 5PM Central Time. John is on the show from 4:30 to 5. You can listen to archives of the show on the web anytime.

#### Books by John Robison:

The Slot Expert's Guide to Playing Slots