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 2. 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 The conditional probability of being dealt two aces is the product of the two probabilities: (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. This article uses both of these approaches.The probability of being dealt various starting hands can be explicitly calculated. In Texas Hold 'em, a player is dealt two down (or hole or pocket) cards. The first card can be any one of 52 playing cards in the deck and the second card can be any one of the 51 remaining cards. This gives 52 × 51 ÷ 2 = 1,326 possible starting hand combinations. (Since the order of the cards is not significant, the 2,652 permutations are divided by the 2 ways of ordering two cards.) Alternatively, the number of possible starting hands is represented as the binomial coefficient
which is the number of possible combinations of choosing 2 cards from a deck of 52 playing cards.
The 1,326 starting hands can be reduced for purposes of determining the probability of starting hands for Hold 'em—since suits have no relative value in poker, many of these hands are identical in value before the flop. The only factors determining the strength of a starting hand are the ranks of the cards and whether the cards share the same suit. Of the 1,326 combinations, there are 169 distinct starting hands grouped into three shapes: 13 pocket pairs (paired hole cards), 13 × 12 ÷ 2 = 78 suited hands and 78 unsuited hands; 13 + 78 + 78 = 169. The relative probability of being dealt a hand of each given shape is different. The following shows the probabilities and odds of being dealt each type of starting hand.