# Texas Hold'em

This approximation gives roughly accurate probabilities up to about 12 outs after the flop, with an absolute average error of 0. One of the 6 ranks is chosen for the pair and two of the four cards in that rank are chosen. Starting hands In Texas Hold'em, a player is dealt two down card or pocket cards. The small blind may also raise the big blind. One interesting circumstance concerns pocket pairs.

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To understand these derivations, the reader should be familiar with the basic properties of the binomial coefficients and their interpretation as the number of ways of choosing elements from a given set. In some popular variations of poker, a player uses the best five-card poker hand out of seven cards. The frequencies are calculated in a manner similar to that shown for 5-card hands, except additional complications arise due to the extra two cards in the 7-card poker hand.

It is notable that the probability of a no-pair hand is less than the probability of a one-pair or two-pair hand. Since suits have no relative value in poker, two hands can be considered identical if one hand can be transformed into the other by swapping suits. Eliminating identical hands that ignore relative suit values leaves 6,, distinct 7-card hands.

The number of distinct 5-card poker hands that are possible from 7 cards is 4, Perhaps surprisingly, this is less than the number of 5-card poker hands from 5 cards because some 5-card hands are impossible with 7 cards e. The following computations show how the above frequencies for 7-card poker hands were determined. Some variants of poker, called lowball , use a low hand to determine the winning hand. In most variants of lowball, the ace is counted as the lowest card and straights and flushes don't count against a low hand, so the lowest hand is the five-high hand A , also called a wheel.

As can be seen from the table, just over half the time a player gets a hand that has no pairs or three- or four-of-a-kind. If aces are not low, simply rotate the hand descriptions so that 6-high replaces 5-high for the best hand and ace-high replaces king-high as the worst hand. The following computations show how the above frequencies for 5-card lowball poker hands were determined. The probability for any specific low hand with 5 distinct ranks i.

The frequency of a 5-high hand or any a specific low hand is calculated by making 5 independent choices for the suit for each rank, which is:. To determine the number of distinct six-high hands, once the six is chosen, the other 4 ranks are chosen from the 5 ranks A to 5 , which is:. This can be generalized for any non-paired low hand.

In some variants of poker a player uses the best five-card low hand made from seven cards. As can be seen from the table, The following computations show how the above frequencies for 7-card lowball poker hands were determined.

To make a low hand of a specific rank four ranks are chosen that are lower than the high rank. There are then three different ways to choose the remaining two cards that are not used in the low hand. Each of these cases must be considered separately:. The suits can be assigned by making 7 independent choices for the suit for each rank, so the number of ways to make a low hand with two distinct higher ranks is:.

In this type of hand there are 6 distinct ranks and one pair. That's just over a third. Don't play any two cards just because they're suited. It only improves your hand by 2. The chance of one of your hole cards making a pair on the flop is If you flop an open-ended straight draw this gives you eight outs eight possible cards that will complete the hand , so you'll hit your hand by the river Just make sure you're getting pot odds the value of the pot versus the value of your bet to see the next card.

If you already have a pair, the odds of flopping a set three of a kind are only 7. If you need two exact cards on the turn and river, the chance of getting them both is only 0.

If you get one, there's a 4. A pair against two overcards is often called a coin-flip or race, because they each win about half the time. People talk about middle suited connectors being better than aces, because of the straight and flush possibilities. There's a reason for folding low hands. If you do get two pair on the flop, the chances of making at least a full house by the river is But if you flop three-of-a-kind, the odds of making a full house or better by the river go up to You bluffed with random cards to steal the blinds and were called by A-K.