Poker Hand Probabilities

Later, the method used for calculating probabilities of initial hands is explained. For the moment, it is useful just to take the numbers from a table of frequencies (probabilities) and cumulative frequencies, like the following one, for hands in draw poker.

This table describes the absolute possibility of receiving 5 cards out of 52 that have the requisite value. The royal flush is just a subset of the straight flush. Out of 40 possible straight flushes, only 4 (10% of them) are royals, so the probability of being dealt a royal is one in about a million and a half hands (0.00015%).

One important item of information from this table is how frequently one can expect to be dealt a pair. The chances of being dealt a pair or better are just at about 50%. The chances of being dealt something stronger than a single pair (the cumulative frequencies for two pair or better) are less than 8%. If the player's criterion for opening a betting round is "jacks or better," by interpolation of this table it can be seen that the chances of meeting that threshold are just a little better than one out of five hands. Many inexperienced players, even employing specific "standards" for when to play cards and when to throw them away, do not realize that only one hand in five is statistically expected as an opening hand under the jacks-or-better standard. Roughly speaking, a player can expect to be dealt two pair about every 20 hands, three of a kind about once in 50 hands, and stronger cards than that about once in every 130 hands or so. To play a hand in response to someone else's bet may not require the same strength as required to open a betting round, but even so, a playable hand is still somewhat infrequent. A pair of sevens or better still only occurs less than once in three hands.

The following table covers the odds of being dealt a specific card or cards in three cards of a deal in stud poker or in two cards of a deal in Texas Hold'em. The methodology is the same as for five-card draw, it is just that only two or three cards are dealt instead of five.

In Texas Hold’em it is important to have high cards in the hole, even if they are not pairs. The odds of being dealt two suited high cards other than a pair are 3.02% and the chance for unsuited high cards is 9.05%. Other openers for Texas Hold’em are listed in the table when it is repeated in the section on that specific game.