32 ways to play the hand

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We continue to march on to National Video Poker Day on Sept. 6 (9/6).

Last week, I talked about how for every deal there are 32 ways to play the hand, ranging from holding all five cards to discarding all five. While most decisions are obvious, many are far less so. Determining which of these 32 ways is the best involves looking at every possible outcome for the draw.

If a player holds four cards, there are 47 possible outcomes – as there are 47 remaining cards in the deck. If the player holds three cards, then there are 1,081 possible outcomes as there are this many ways to draw two cards from the remaining 47 (aka 47 choose 2).

While some might be able to quickly figure all the possible outcomes when drawing only one card, it gets much more difficult with two cards and near impossible when drawing three, four or five cards. That’s why we use computer programs to do all the heavy lifting. With computers, we can literally look at every possible draw for each possible way to play a hand and determine every possible outcome.

It then calculates what is called the expected value of the hand or the average coins we can expect to have returned if we play the hand that way. To calculate it we take the payouts of all the possible outcomes and divide it by the number of possible outcomes. Let’s look at a simple example.

If a player is dealt Two Pair – 5’s and 7’s and a Jack, I think it is fairly obvious he will hold the Two Pair. But, what is the expected value of this hand? We look at the 47 possible draws and realize four will result in a Full House and the other 43 will leave the hand as a Two Pair. So the payout is 4 x 9 (for the 4 Full Houses) plus 43 x 2 (for the 43 Two Pairs). This totals 122. There are 47 possible outcomes, so the expected value is 122/47 or 2.60.

We divide by the possible outcomes because we are frequently comparing hands with different numbers of possible outcomes to decide how to play a hand. For example, a player might be dealt a Low Pair that is also a 4-Card Flush. For example: 5-hearts, 7-hearts, 10-hearts, K-hearts, 5-diamonds.

While our program looks at all 32 ways to play the hand, it is readily apparent there are two “serious” potential ways to play it. You can hold the Low Pair or you can hold the 4-Card Flush. With the 4-Card Flush, there will be 47 possible outcomes resulting in a Flush and three in a High Pair. This gives us 57 units returned; 57/47 = 1.21.

With the Low Pair there are 16,215 possible outcomes. Obviously, if we looked at only units returned, it would clearly win, but we’d be comparing apples and oranges. We need to divide by the number of possible outcomes to make sure we are comparing the expected values properly.

In the case of our Low Pair, the possible outcomes are 45 Quads, 165 Full Houses, 1,854 Trips, 2,592 Two Pairs and the remaining 11,559 are losing hands. When we add up the payouts, we get 13,356 units. We divided this by the 16,215 possible outcomes to get an expected value of 0.82.

So, the 4-Card Flush (with one High Card) has an expected value of 1.21 and the Low Pair has an expected value of 0.81. We now know the better way to play the hand and the decision is not even particularly close. The 4-Card Flush has an expected value roughly 50% greater than the Low Pair. This hand is extremely common and if you have been playing it wrong, correcting just this should make a world of difference to how you do going forward.

What if our 4-Card Flush with one High Card had been a 4-Card Straight with one High Card? The Straight only pays four and there are only eight ways to make it. All of a sudden, our payout is now (4 x 8) + (3 x 1) or 35. And 35/47 is about 0.74 so our decision just took a drastic turn. Now the Low Pair has an expected value higher than the alternative 4-Card Straight.

I’m guessing many players when they first start playing video poker – and do so without having learned the strategy – play these two hands the same way: either always go for 4-Card Straight/Flush or always go for Low Pair. In reality, neither is correct. We play a 4-Card Flush over a Low Pair, which is played over a 4-Card Straight.

We’ve just scratched the surface of the strategy for video poker and already I’ve hopefully made many of you better players!

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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