The card game Texas Hold’em Poker is perfect for demonstrating how to calculate probabilities. I will assume that the reader already knows the basic rules of the game. First of all I will give an easy to remember model of what a probability is:

Think of a box with black and white balls in it. We draw one ball from the box. The probability of drawing a white ball equals the number of white balls divided by the overall number of all balls in the box.

If we don’t know the number of balls in the box, we can (theoretically) still calculate the probability of drawing a white ball by repeatedly drawing a bill and counting how many white balls we get. Then we can estimate the probability by dividing the number of white balls by the number of times we drew a ball. As more repetitions we make, the more exact our estimation will be.

You can model any stochastic question with balls in box and variations of it, eg. drawing balls multiple times with or without putting the drawn ball back in the box.

Lets apply this model to poker. Of course, here we draw cards from a stack. Once a card is drawn, it is taken and will never be drawn again. In the words of the original model this means: drawing without putting balls back into the box.

There are 52 possibilities to draw one first card. Then there remain 51 possibilities to draw a second cards. So there are 52*51 possible starting hands for a player in Texas Hold’em.

The most often cited calculation in Poker are Pot Odds. Pot Odds are a rule of thumb how you should act after the flop, when you hold a draw and another player has already raised before you. Mathematically this situation can be seen as a bet. The stake is the amount of chips you have to call and the possible earning is the overall pot (including all bets). Your Pot Odds are the fraction of these two.

An 'Out ' is a card that completes your draw. The possibility of completing your draw (and hopefully winning the showdown) **with the turn or river** can be estimated as `4 * number of outs`

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bet to call --------------- <=> 4 * Outs pot

If the right side is larger then you can call and hope to complete your draw.

When you see a Poker tournament on TV there are numbers next to each players hands. These numbers represent the probability of this hand to win against all other players, for each possible combination of remaining cards. Of course you can only calculate these probabilities, if you know the other players cards. In contrast to that, I was looking for probabilities that tell me how to act in a given situation within a game, knowing only my cards and the cards on the board.

Given are my two hole cards and the cards on the board (if any). Now I could calculate all possible combinations of cards for all opponents. But there are far too many possibilities to get through all of them in a lifes time. So I try to narrow the field by simulating how the other players might act.