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# Poker probability for Texas hold ’em

In poker, the probability of many events can be determined by direct calculation. This article discusses how to compute the probabilities for many commonly occurring events in the game of Texas hold ’em and provides some probabilities and odds[1] for specific situations. In most cases, the probabilities and odds are approximations due to rounding.When calculating probabilities for a card game such as Texas Hold ’em, there are two basic approaches.

1. Determine the number of outcomes that satisfy the condition being evaluated and divide this by the total number of possible outcomes. For example, there are six outcomes (ignoring order) for being dealt a pair of aces in Hold’ em: {A♠, A♥}, {A♠, A♦}, {A♠, A♣}, {A♥, A♦}, {A♥, A♣}, and {A♦, A♣}. There are 52 ways to pick the first card and 51 ways to pick the second card and two ways to order the two cards yielding 52 × 51 ÷ 2 = 1,326 possible outcomes of being dealt two cards (also ignoring order). This gives a probability of being dealt two aces of 6/1326 = 1/221.
2. Use conditional probabilities, or in more complex situations, a decision graph. There are 4 ways to be dealt an ace out of 52 choices for the first card resulting in a probability of 4/52 = 1/13. There are 3 ways of getting dealt an ace out of 51 choices on the second card after being dealt an ace on the first card for a probability of 3/51 = 1/17. The conditional probability of getting dealt two aces is the product of the two probabilities: (1/13)x(1/17) = 1/221. (Note that in this case the total is not divided by 2 ways of ordering the cards because both cards must be an ace—reordering would still require the first and second cards to be an ace, so there is only one way to order the two cards.)

Often, the key to determining probability is selecting the best approach for a given problem.

## Notes

1. ^  The odds presented in this article use the notation x : 1 which translates to x to 1 odds against the event happening. The odds are calculated from the probability p of the event happening using the formula: odds = [(1 − p) ÷ p] : 1, or odds = [(1 ÷ p) − 1] : 1. Another way of expressing the odds x : 1 is to state that there is a 1 in x+1 chance of the event occurring or the probability of the event occurring is 1/(x+1). So for example, the odds of a role of a fair six-sided die coming up three is 5 : 1 against because there are 5 chances for a number other than three and 1 chance for a three; alternatively, this could be described as a 1 in 6 chance or $\begin{matrix}\frac{1}{6}\end{matrix}$ probability of a three being rolled because the three is 1 of 6 equally-likely possible outcomes.

## References

• Mike Petriv (1996). Hold’em Odds Book. Objective Observer Press. ISBN 0-968122-302.
• King Yao (2005). Weighing the Odds in Hold ’em Poker. Pi Yee Press. ISBN 0-935926-25-9.
• Dan Harrington, Bill Robertie (2005). Harrington on Hold’em Volume 1: Strategic Play. Two Plus Two Publishing. ISBN 1-880685-33-7.