There are several important math concepts to understand in poker. I’m going to discuss some of the more applicable math concepts I’ve noticed misunderstood by a lot of players. You may already be familiar with most of these terms, but hopefully this simplifies it in an easily digestible format. My goal is that you can walk away from this chapter and fully understand these concepts. Even if you had some idea and understanding of them before, they should be super clear after you read this chapter.
Counting card combinations at the table in most situations will be nearly impossible. However, knowing and understanding about how many combinations of hands exists for specific hands that you beat, and have you beat will help in the decision-making process. In real time you will rarely be exact with this count, but you can get close, and in post analysis it’s helpful to know in analyzing and reflecting on your play.
Total possible hand combinations for any two card hand are multipliers of the cards left in the deck. So pre-flop, there are 16 total combinations of AK. Four suites of aces, and four suites of kings, or 4 x 4 = 16. Anytime you remove a card from the deck, you remove one possible combination of that particular hand being possible. If you saw a flop of Qs9d4h, and you wanted to know how many combinations of AQo someone could have, then we remove the Qs from counting. We’d come up with 4 suites of aces times 3 remaining suites of queens, or 4 x 3 = 12.
Most really good combinational analysis will come post analysis after playing the hand. However, there are some situations where knowing common possible combinations that beat you can be helpful in the decision making. If you have AdTc on a flop of AsQdTh, you’ll know that there are only 5 combinations of sets, 16 combinations of a flopped straight, and 6 combinations of a higher two pair (remove the Ad) that beat you for a total of 27 combinations. There are only 9 combinations of worse two pairs that you beat, and 4 that you split with. So if you bet and are raised on the flop, you can fold in most situations. Although we’d discount some of the hand ranges such as a flopped straight that might look to just call, there are still a lot of hands that have you beat, even if you add a good amount of air hands to your opponent’s range.
Another helpful aspect of understanding combinations is applied to pre-flop hand ranges. If you’re facing a tight pre-flop 3-bettor, say someone who 3-bets only slightly over 2% of their hand range, we’d safely say they are 3-betting KK+, AKs, and AKo only. If we took those four hands, it might seem as if they are 3-betting each about 25% of the time. However, if we look at the total combinations of hands, they are actually 3-betting KK+ about 43% of the time (6 combinations twice for a total of 12), and AKo, AKs the remaining 57% of the time (12 offsuit combinations plus 4 suited combinations for a total of 16). Most of the time they’re going to have AK instead of AA or KK.
Probability is the estimate that something will become true or occur. It doesn’t however ensure that an event or outcome will happen though. Since each event is not related to the prior event, probability theory states that there is X% occurrence for Y to happen, but it cannot say that Y will ever happen if there’s even a slight chance that it won’t. This sometimes confuses gamblers when probability is applied to expected value (more on this later).
If you’re running bad in poker, you are not guaranteed that you will not run bad for very long extended time periods. This is what tends to lead many poker players, and gamblers in general, to ruin or go bust. They begin to believe they are “due” to win because their luck has been so bad, but this is not the case. If the probability of Y event happening is consistently in your favor, then the odds of Y event will increase, but it also could never happen.
There are a few popular probability theories, and one that is used quite often in poker is Bayesian probability. The reason it’s commonly used and applied is because Bayesian probability theory takes into account prior information to help predict future outcomes. This is highly applicable to poker situations since in many situations we’ll have information about our opponents, their tendencies, and in some case HUD stats which will help create a better probability that your opponent has Y hand given this prior information. Using Bayesian probability in situations like this allows for a more accurate prediction of the probability of many situations in poker.
Equity is the percentage of the pot that belongs to you based on the percentage of the time you’ll win the pot over many large samples. This is not the same as expected value, although the two are sometimes interchanged in poker conversations often. Equity is purely the percentage of the time you’ll win, lose or split the pot.
For example, if you saw a flop of: QcTd6h which had $100 already in the pot, and you held AsKh and your opponent had KK, and then went all- in and you called, you’d have ~28% equity in the pot. ~28% of the time on average you could expect to win this pot. Whether this was a good play to call or not would depend on our equity, in conjunction with our expected value.
Expected Value (EV)
Expected value is the average money won or lost based on your equity and odds of the call or raise you are making. Using the above example, if our opponent bet $60 on the flop, and we called, we’d use the current equity we have in the pot to determine our expected value. In this case 28% of the time we win $160, and 72% of the time we lose $60.
(160(.28)) – (60(.72)) = +1.6
So calling the all-in on the flop will net a positive expected value of $1.6 on average. Our equity is low in this situation, so a majority of the time we will lose, but based on the money in the pot, and our opponent’s bet, the expected value net is positive.
Fold equity is the equity gained in your hand if you bet or raise and get your opponent to fold. It’s an estimation of how often you believe your opponent will fold multiplied by your opponent’s current equity. So if you opponent has 75% equity in their hand, but you estimate that if you bet, you can get your opponent to fold 50% of the time, then your fold equity would be:
75% x 50% = 37.5%
So your new hand equity if you bet in this situation would be 25% (your hand equity if you both saw all five cards) + 37.5% = 62.5%. You’ve increased the equity in your hand from 25% to 62.5%, just by betting. This is why aggression pays off so often in poker because every time you bet, you give your opponent the chance to completely forfeit their equity in the hand. Opponents are rarely drawing completely dead, so when you can get them to fold when they have an equity advantage, it’s a huge +EV situation for you.