Expected Value differs from game to game

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Last week, I went through the process of how an Expected Value is calculated. Specifically, I showed the expected value of a dealt Three of a Kind for a full-pay Jacks or Better video Poker.

I alluded to the fact that the expected value of a Three of a Kind is irrelevant for Jacks or Better but quite important for a Double Double Bonus game. First of all, it should come as no surprise that the expected value for Trips will be different for Double Double. Further, since there are different pays for different Quads and yet different pays for those Quads with kickers, we will have different expected values depending on what the rank of the Trips we have is.

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There is no difference in the payout of Trips or Full House between Jacks or Better and Double Double. So, all of the impact is due to the different payouts of the Four of a Kinds. For 5’s through K’s, there is no additional pay for kickers. We just use 50 for the payout of all potential Four of a Kinds when we start with three 5’s through K’s. As a result, the expected value goes to 5.4 from 4.3 for Jacks or Better.

The expected value for 2’s through 4’s goes all the way to 7+ when we take into account the 80 payouts without kickers or 160 with a kicker. Three Aces has an expected value of just below 12 to just below 13 depending on whether or not we have to discard a kicker (or two) to go for the Four of a Kind.

If we are dealt just a Three of a Kind, there isn’t much of a decision and the specific expected values are still irrelevant. But, what if we are dealt a Full House? What is the right play then?

Well, with a payout of 9 for a Full House, the expected value is rather easy to calculate — it is 9. So, now we need to decide what the right play is. If we have three 7’s and 2 10’s, do we keep the Full House or go for the Four of a Kind? The expected value of keeping the Trips is around 5.4. It is actually reduced a little from the true number because we will be discarding a Pair. But at 5.4 the answer is pretty straight forward.

So, if the true number is a little lower it doesn’t matter. We keep the Full House.

What if our Full House is three 2’s and two 9’s? Well, the expected value is 7+, but this is still a long ways from the expected value of 9 of the Full House. So, again, we keep the Full House.

But what if the Full House is three Aces and two Jacks? Now, the expected value is nearly 13. Throwing away a pair will reduce our expected value a tiny bit, but will not bring it down to anywhere 9. Even if we throw out a pair of 2’s, it will still be well above the expected value of the Full House.

Thus, we discard the Pair and go for the Four Aces. Only 46 out 1,081 will result in the Quads. Of these, up to 12 will result in hitting it with the kicker. But, these will result in upping the payout to 160 or 400. This is well worth the risk of taking your payout back down to the Trips level from your Full House.

From all of this, there are many lessons to be learned. One is that the expected value of any particular hand varies depending on the specific paytable. With a payout of only 1 for Two Pair, all of the hands near the bottom of our Strategy Table have very different expected values.

Another is that to properly play any game you need to learn the Strategy Table for that game. If you take the Jacks or Better strategy over to a Double Double Bonus game, you’ll be hurting yourself. 

About the Author

Elliot Frome

Elliot Frome’s roots run deep into gaming theory and analysis. His father, Lenny, was a pioneer in developing video poker strategy in the 1980s and is credited with raising its popularity to dizzying heights. Elliot is a second generation gaming author and analyst with nearly 20 years of programming experience.

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