Poker Probabilities

Experienced poker players do not need to get out a calculator every time a decision point occurs in a game. That does not mean, however, that calculations are unimportant. Every risk in poker involves comparing the "real odds" to the "pot odds." This is just a compact way of saying something that is true for every intelligent bet, no matter what the context: The cost of the bet divided by the payoff for winning (i.e., the financial proposition implicit in the betting situation) needs to be equal to or better than the "true odds" of the outcome, that is the expectation in the "real world."

To cement this in the mind: If the pot has $50 and the cost to keep in the game is $10, the payoff (assuming no further play) will be $60, and the cost will have been $10. This ratio yields 16.67%. If the "true" probability is greater than that number, say, 25%, then the bet makes sense. There are other ways to arrange these three variables and come to the same conclusion. For example, "pot odds" are 5:1. If "true odds" are 3:1 (as in the case of a 25% chance of success), then the pot odds will pay off more handsomely than the real risk involved. The essential aspects are simply (1) don't get confused and (2) don't make stupid bets - that is, bets where you are overpaying for the chance to win.

Of course, knowing what the physical expectations or "true odds" really are is not always straightforward. This page provides the theoretical underpinning for calculating just about any probability in a poker situation. People who are phobic about math and statistics can still derive some benefit from the discussion, as even a "general idea" about these matters can help sharpen the intuition when added to a good dose of experience.

The first and most detailed part of the generic discussion of poker probabilities uses five-card draw poker as the model. Later, stud and Hold'em are added in. The probabilities come from two basic kinds of questions:

1. What are my odds of being dealt a _____ (fill in the blank with pair, trip, flush, etc.)? and

2. What are my odds, given that I have a _____ (pair, trip, etc.) of improving my hand with a ____ (two, three, etc.) card draw?

The first question addresses the initial deal. The technique employed to calculate the initial deal can equally be employed to calculate the odds of receiving, say, a pair in the initial deal of a game of stud or Hold'em as well.

The second question is about "conditional probabilities": Now that you have [whatever it is you have], how likely is it that the next card [or a draw of two or more cards] will have the card or cards you want? The theory is the same, obviously, for draw, stud and community card games, but the techniques will vary a little.