Some hands more complicated

Last week I explained how an expected value is calculated for video poker. I used an overly simplistic example of a Three of a Kind.

You don’t really need to know the expected value of a Three of a Kind as there is no game I can think of where you would do anything but hold that hand.

But not all hands are this obvious. What if you are dealt the following?

As I described last week, our computer program will calculate the expected value for all 32 possible ways to play this hand, including holding the 4 of Clubs/10 of Diamonds which would be absolutely awful. Any decent player will quickly realize that there are likely three ways to play this hand. The player can hold the Low Pair, the 3-Card Royal or the 4-Card Flush.

We go to our program to calculate the expected values and we get these results. The Low Pair has an expected value of 0.82. The 3-Card Royal has an expected value of 1.47. The 4-Card Flush has an expected value of 1.15.

To remind everyone, the expected value is calculated by looking at every possible draw and summing up the payouts of the final hands and dividing this sum by the total number of possible draws. So, for a 3-Card Royal, the final hands will be 1 Royal, 2 Straight Flushes, 33 Flushes, 45 Straights, 7 Three of a Kinds, 21 Two Pairs and 246 High Pairs. The rest of the hands will be losses.

If we add up these totals we get (1 x 800) + (2 x 50) + (33 x 6) + (45 x 4) + (7 x 3) + (21 x 2) + (246 x 1) = 1587. With 2 cards drawn there are 1081 possible draws. So, 1587 / 1081 = 1.47.

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A similar calculation is performed for the other 31 possible ways to play the hand including our other two plausible ones. This is not an approximation or a gut feel. This is simply math telling us how we can expect to do over the long run. It does not predict how we will do on our specific draw.

The next two cards out might be the other two 10’s. If we go for the 4-Card Flush, we will have a losing hand. If we go for the 3-Card Royal, we’ll have a Three of a Kind. If we go for the Low Pair, we’ll wind up with Quads. In any given draw, lightning may strike, but that does not mean it was the right play. 

It is also important to remember that little changes to the hand can impact the expected value. For instance, if that 10 of Diamonds was in fact the Jack of Diamonds, then it is a High Pair instead of a Low Pair. The High Pair has an expected value of 1.54 which is greater than that of a 3-Card Royal.

So, in the simplest terms, you play a High Pair over a 3-Card Royal over a Low Pair. It might be exciting to go for that Royal, but it is literally a 1-in-1,000+ shot. Better to take the sure winner if you’re interested in winning in the long run.