Poker Probability (Texas Hold 'em)
In poker, the probability of many events can be determined by direct calculation. This article discusses computing probabilities for many commonly occurring events in the game of Texas hold 'em and provides some probabilities and odds 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. The first approach is to 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 possible outcomes when being dealt two cards (also ignoring order). This gives a probability of being dealt two aces of .
The second approach is to use conditional probabilities, or in more complex situations, a decision tree. There are 4 ways to be dealt an ace out of 52 choices for the first card resulting in a probability of 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 This value is the conditional probability that second card dealt is an ace given the condition that the first card dealt is an ace. The joint probability of being dealt two aces is the product of the two probabilities:
Often, the key to determining probability is selecting the best approach for a given problem. This article uses both of these approaches.
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