# Mathematics in poker

How can we use simple mathematical concepts and operations to improve our decision making in poker? In this article we will show you how to profit at the poker tables using mathematics. The fact is that the best poker players use mathematics constantly in their decision making.

We hope that these tips and observations will be useful both for your offline poker career and especially if you play poker online; because the practice of poker on the internet usually involves a much greater number of hands played and it is there, when we have samples of thousands of hands where probabilities, statistics and basically mathematics in poker are an essential element.

## A practical example: pot odds

Mathematics helps us in all facets of the game. Let’s see how mathematics helps us in the subject of pot odds as we will see in the following two particular cases:

### – Case 1

Let’s say for example that there is a pot of \$10 and when we reach the river (last community card) we are faced with a bet of \$5. Instinct or intuition can play tricks on us by making us believe that we need to win 50% of the time for the call to be good in the long run. But in reality, we wouldn’t even need that 50% for the call to be good. So we don’t need to have a full house or a great hand to see that bet.

## Odds at the poker table

The more players at the table, the more important it is to keep statistics and odds in mind.

Let’s imagine we only have to bet \$1 to win a pot of \$1,000,000 and we have a slim chance of winning the pot. Even if we have a remote chance, we should still pay out. Whether those odds are one in a hundred or one in ten thousand, it seems pretty obvious that the huge winnings outweigh the small costs. Let’s go back to our particular case.

How often will I have to pay that \$5 to make it mathematically profitable? Just apply a simple formula. You don’t need to be an expert in game theory or know concepts such as GTO poker.

Proportion that this bet would represent for the pot (in %) = Frequency that I should win the hand (in %).

In other words, for this particular case, when we hit the river there was a pot of \$10 and the opponent bets us \$5. In other words, we need to pay \$5 for a total pot of \$15. This is equivalent to paying 1/3 of the pot, so we need to be successful one out of three times for the call to be a correct decision.

In other words, we have 33.33% of the time we will need to win the hand for it to be a good call.

### – Case 2:

We reach the river in a poker hand and there is a pot of \$100 but we don’t manage to call any moves and decide to make a bet of \$50 to try to bluff our way into the pot. How often should our opponent fold to make our action profitable in the long run? Again, our intuition may give us the wrong information and we may believe that we need a 50% success rate for our move to be profitable, but again we would be wrong.

But we must use similar reasoning to the previous case to calculate the probability of success of our bluff. In ratio, we have to rely on this mathematical principle:

## The amount bluff bet / total pot after the bluff bet.

In this case: 50 /150 = 1/3; therefore, our opponent should fold one out of three times for our move to be at least neutral and not lose money. From that frequency of withdrawals onwards, it would be a profitable move. That is, if we assume that our opponent is going to withdraw 30, 25, 20%, our move would be profitable.

But if, on the other hand, we believe that 35, 40, 50% of the time our opponent will pay that bet, the profitable thing to do is not to bluff.

In percentage: 1/3 = 33.33%.

So that is the minimum fold percentage we need for our move not to be a losing one. Note that the more the opponent tends to fold, the less money we should risk on the bluff bet. Although it may seem shocking, we can lose more than half of the hands in such situations and our move will be positive in the long run.

But we don’t want to mislead you, and the decisions to be made are often much more complicated and it is also difficult to estimate the odds that we think our player is going to fold to our bluff, or similarly, when we are in reverse situations, it is not so easy to estimate the odds that our opponent is bluffing.

We all know that in poker schools, mathematics, statistics and probabilities are a fundamental subject. But we want to make you see that in poker, mathematics can make the difference between a mediocre, losing player and a brilliant, winning player.