Poker Lesson 16: Using Mathematics

Early human civilization had no concept for zero. Back then the work was hard, and there was also the possibility of being eaten by a giant ground sloth, but in general folks were happier. They also were not very good poker players. With the advent of zero came the birth of mathematics. With math we got the industrial revolution, nuclear weapons and Microsoft. People are less happy today and tend to fling themselves off tall buildings more often, but we are better poker players.

How does a player use math to figure out what cards their opponent is holding? Well you can’t, not absolutely. You can use your skills with numbers to determine the probability of what might be sitting in their hand. The exact science of math, in this case, works with the subjective practice of observation. Before you can break down the probability of what they have in their hand, you must first use what you have learned about competitor’s tendencies in opening, calling and raising.

A good example of this is that if you know that an opponent will play more aggressively with a three deuces or higher before the draw, you can fall back on math to determine what hand will most likely beat them. If your opponent raises, obviously calling while holding a triplet of threes is not worth the effort on the off chance that they have exactly three twos. Based on what you know of the opponent and what is in your hand, there are simply too many ways you can be beaten. If you have something like three fives or sixes though, the pot odds come into play and make calling the right thing to do. Your chances of improving on the draw have increased and the odds of you getting a full house or four-of-a-kind are better. It will also take a hand of three sevens or better to defeat your cards.

When calculating your chances, sometimes using Bayes’ Theorem can be of help. Baye, a gentleman who probably wore flood pants and understood the functional beauty of a pocket protector, came up with a mathematical formula for calculating conditional probability. I won’t bore you with the actual numbers, seeing as Baye did not go to the sort of parties where poker was a factor, so the game has little to do with the construction of his theorem.

Once you have worked out the hands your opponent is likely to bet on in certain situations, you determine the probability that it is one of those hands they are actually holding. For example, if you know that your opponent will open in draw poker with a three-of-a-kind or two pair but will not open with one pair, and will check (as a slowplay) when they are holding cards they don’t need to improve, than the odds are 5-to-2 against that player’s having a three-of-a-kind when they do open. How did we come to this determination? Easy, according to draw poker distribution a player will be dealt two pair 5% of the time and triplets 2% of the time. By simply comparing the two percentages, you arrive at a ratio of 5-to-2. In short, your opponent probably has a two pair.

This works for Hold’em as well. Let’s say your opponent raises big before the flop. Your observations tell you this only happens when they have premium starting cards such as a pair of Kings, two Aces, or an AceKing combo. The probability that the player get dealt a pair of aces or king as a starting hand is .45%. In short, .9% of the time your opponent will be holding a pair of Kings or Aces. Also, the other starting hand your observations have told you that they are willing to raise on, the Ace, King combo, happens 1.2% of the time. Taking these two calculations into account with what you know about the other player and you have chances that are 4-to-3 in favor of your opponent holding an Ace/King combo instead of a pocket pair of Aces or Kings. Even in the best case scenario of the Ace/King combo, you still should not call that raise if you are holding a pair of Queens or below. You hold a slight advantage in the case of the Ace/King combo…an advantage that is precarious at best considering the outs sitting in the deck, but you lose head to head if they truly are holding a pair of Aces or Kings.

Here are some other percentages that may help you make decisions during the hand:

- Hitting a flush draw (both turn and river, needing one card to hit): 35%
- Hitting an open-ended straight draw (i.e. 4 straight cards, need one on either end to hit on turn or river): 31.5%
- Hitting a gut-shot draw on turn or river: 16.5%
- Being dealt a pocket pair: 5.88%
- Being dealt suited cards: 23.5%
- Hitting a three of a kind or quads at the flop when you hold a pocket pair: 11.8%
- Making a pair at the flop, holding two unpaired cards in the hole: 32.4%
- Being dealt AA or KK: 0.45%

Remember, these numbers are only valuable if you have some idea of what your opponent will do in a given situation. Subjective observation and cold objective numbers work hand in hand to help you make the right call.