Mathematics of Poker

Introduction to Probability and Statistics

In Hold'em poker, mathematical poker analyzes are based on theories of probability and statistical computation. Probability and statistics are not the same topics, although they are directly related. Probability is applied to the description of the theoretical results from random events. Statistics is applied to the description of actual, observed results from random events.

There are different other ways to distinguish between probability and statistics and the selected one is somewhat subtle, but it works well enough for other purposes. Probability is about theoretical results whereas statistics is about texas holdem observations results.


A "probability" is a number between zero and one, which represents the relative frequency of a particular result occurring as a result of a random event. A probability of zero means the result cannot occur. A probability means the result will occur with certainty. A probability of 0.5 means the event will occur one half of the time.

For example, shuffle a deck of holdem overcard and turn over the top card. That's a random event. One of possible results is that the card is black, either a Spade or a Club. The probability of this particular result is 0.5. It means that one half of the time the card you draw will be one of the black suits.

A "probability distribution " is a list of all possible, mutually exclusive results and their related probabilities. Here by "mutually exclusive " means that the results don't go beyond in some way. Each result is defined is such a manner as to be different from other results. For example, considering our event of shuffling the deck and turning over the top card, one probability distribution we could define would be:

Black Card 0.5
Red Card 0.5

Law of Averages

The "law of averages" is a limit theorem that says the sample mean gets randomly close to the actual mean as the sample gets bigger. "Mean " is technical term for average. "Actual mean " is the theoretical average that's implied by a probability distribution: "sample mean " is about actual observed results. A sample mean is the poker mathematics average of some actual result.

For example, in our black-card red-card probability distribution, think about a black card as representing value 1 (you win dollar if the card turned up is black) and the red card is the value 0 (zero). The mean of this distribution is 0.5. On the average you win $.50 for every turn of the card. Half the time the card is black and you get $1 and half the time the card is red and you get nothing.

If the deck is shuffled and turned over a card ten times, on an average we would get a black card five times, but we might get six black cards or four black cards and in some unusual results, we might even get a black card all ten times. Let's think that happens, we shuffle and turn card ten times, turning over a black card every time. What does the law of averages tell us about the results from the next ten times we turn a free holdem card or the next 100 times or the next 1000 times?

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