Poker maths

Most of the hardcore calculations I do are away from the table. Then I take the results and try to form some general rules or simpler procedures for estimating my odds.

I’ve written several different programs of my own and use them and some programs that others have written, (such as for calculating poker odds and solving simplified poker games. I’ve done enough calculations of pre-flop situations that I don’t have to do much calculation. For example, if it’s heads up I know that I have odds to call an all-in raise or re-raise with practically any Ace if I’m getting pot odds of better than 3:1; even if my opponent will only make that play with QQ, KK, or AA. (“Pot odds” is the size of the pot divided by the size of the bet that has to be called. For example, suppose the pot is £1000, my opponent bets all-in for £500 more. The pot is now £1500, and I have to call £500, so the pot odds are 3:1. If I have a 25% chance of winning or better, the correct play is to call).

Well, to be exact, if the pot odds are exactly 3:1, I should throw away A2, A3, and A6 through AQ off suit. But if the pot is just slightly larger, or if my opponent might also have JJ or AK, then I’m supposed to call. In fact, getting exactly 3:1 against an opponent whom you know has either QQ, KK, AA, or AK, you’re getting the right pot odds to call with quite a lot of ugly-looking hands. Any pair. Any AK. Any suited cards except K2s through K9s, Q2s, and J2s. Off-suit 43 through JT, 53 through T8. And off-suit AT, A5, A4, and A3.

At the table, do I calculate exactly what my probability of winning is pre-flop? No. What I do know is that if I raise pre-flop I will almost never fold to a single all-in re-raise when I’m getting 3:1 or even 2:1, if I had something worth stealing with.

On the flop, I know that I’ll win half the time if I have 13 outs (“outs” are all the cards that you win with against a better hand). A good approximation for your chance of winning is 2% per out on the turn and twice that, 4% per out, on the flop. This will usually underestimate your probability of winning, but being a little conservative is OK, especially since one or more of your outs might be in your opponent’s hand or might help him also.

The next step is to think about the hands that your opponent is likely to hold and assigning some rough probabilities to them. Total the chance of winning against each of them weighted by their probabilities. Maybe your opponent is bluffing or semi-bluffing, or maybe you think you’ve picked up a tell that makes you think your opponent has a strong hand — adjust accordingly. You don’t need to do a perfect calculation, just close enough so you’re only making at most a small mistake. Thinking too hard to get a more exact answer will make it harder for you to concentrate on other important things like looking for tells and remembering your opponents’ tendencies.

If you’re playing against a strong opponent, you might want to use game theory to determine whether you should call or fold, especially on the river. If you have a medium-strength hand, game theory says you should call often enough to make your opponent almost indifferent to bluffing with clearly losing hands. If your opponent bets the pot on the river, you should call roughly half the time. If he bets a fraction X of the pot, you should call roughly 1/(X+1) of the time. So ask yourself, is my hand in the top X/(X+1) of the hands I might have in this situation? If it’s close, or if you think your hand is too well-defined by the betting, you can even resort to flipping a coin.

That’s pretty much all the maths you need to do while at the table. It may seem like too much to some, but after a while you should get used to it. If you can’t do it while you’re at the poker table, practice doing it away from the poker table. After a time, most situations will become second nature.