Singing 32 ways to play your hand in video poker

In 1975, Paul Simon sang about 50 Ways to Leave Your Lover. While I like the song, I have to admit, I’ve never listened to it closely enough to know if he actually lists 50 ways or even starts to list any.
Fortunately, in video poker, there are only 32 ways. No, not to leave your lover, but to draw on your hand. When I tell people this, the most common reaction is to give me that little deer in the headlights type of look. So, let me explain.
When dealt an initial 5-card hand in video poker, you have to choose which of the 32 ways to go with it.
What does this 32 represent? Well, I think it is fairly obvious that you can choose to keep the five cards you were dealt. That’s way No. 1. You can also choose to discard all five cards and draw new ones. We’ll call that way No. 2.
If you decide you want to draw 1 card, there are five ways you can do this. You can discard any one of the five cards originally dealt to you. So, these are ways 3-7. In similar fashion, you can choose to keep any one of the five cards and discard the remaining four. This gives us five more ways or 8-12.
Only 20 more ways to go, but it does get a bit more complex. The last remaining choices are to discard two (and keep three) or to discard three (and keep two). If we have five cards and we want to pick two of them, there are 10 combinations we can pick from. Mathematically, this is called 5 Choose 2.
To calculate this, we need to use something called a factorial, which is a number multiplied by itself and all numbers less than it, down to 1. It is represented by an exclamation mark. So five factorial (5!) is 5 x 4 x 3 x 2 x 1. This works out to be 120. 5 Choose 2 is mathematically equivalent to 5! divided by 2! divided by (5-2)!
In the end, this turns out to be 10. This is true so long as order does not matter. Thus we are picking two cards that are not assigned a position of 1 and 2, but are simply those we are choosing to keep or discard. So, we have our remaining 20 ways – 10 that go with discarding two cards and 10 that go with discarding three. This gives us ways 13-32.
Of course, the average player doesn’t play video poker thinking through each of the 32 ways. Instead, the human brain can look at the 5-cards dealt to him and determine what hand type he has and match it to the strategy table.
The player also doesn’t take into account the specific card position to notice that there are 10 ways to discard 2 cards. Instead that person might recognize having a 3-card Straight Flush and needing to discard the other two cards.
For about 75% of the hands, the choice is very easy and straight forward. The importance of strategy comes into play with the remaining 25%. But, even for these, most of the time there are only two ways the player must consider.
Perhaps he has a Low Pair and a 4-card Flush or a 3-card Straight Flush that is also a 4-card Straight. Once in a while there’s a hand that has three ways that must be considered. The hand might be a 3-card Straight Flush, a 4-card Flush and a Low Pair.
I’m sure if I thought about it long enough, I’d find cases where the player might have to consider four ways, but these are rare, if they exist at all.
So, why do we concern ourselves with talking about the 32 different ways a hand can be played?
On a theoretical level, all 32 must be analyzed to determine which maximizes our overall payback. For a computer program, it is almost easier for it to consider every possibility rather than attempt to eliminate them through some form of human assumptions programmed in. These assumptions might lead to errors.
Who knew 20 years ago they would come up with all the bonus pay tables that they have? Who knows what “bizarre” pay table might be offered in the future?
Thus, when I create programs that analyze a game like video poker, I don’t make any assumptions about how the hand should be played. I allow the computer to look over every possible combination.
When the computer is analyzing every draw for all possible ways to play a hand for each of the 2,598,960 possible 5-card deals, it is actually analyzing approximately 6.7 trillion situations. It is almost hard to believe that when all is said and done, we can boil this down to about 35-40 lines on a strategy table.

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