Mathematics of Poker

See also

• Game Theory and Poker (Part I)
• Game Theory and Poker (Part II)

Some useful mathematical facts about poker games:

Holdem:

• If you flop a flush or a straight draw, with no other ways of winning, you are approximately a 2 to 1 underdog.
• If you have a small pair and need to make trips to win, you will not have sufficient pot odds in limit hold'em to take a card off on the turn.
• If you have 13 outs on the flop, you will typically be a slight underdog.
• If you have 14 outs on the flop, you will typically be a slight favorite.
• A big pocket pair against a small pocket pair is slightly more than a 4 to 1 favorite to win the hand.
• If you have a pocket pair, you are around a 7.5 to 1 underdog to flop a set.

Seven-card stud:

• If you have an open-end straight draw or a flush draw in the first four cards, you will typically be a slight underdog to make the hand. Of course, you may win by making something other than the straight or flush, or you may win by playing aggressively.
• A starting three-card straight has about an even chance of winning against an underpair with a small kicker if there is no betting after the first three cards.

Lowball:

• Against a one-card draw, in Ace-to-Five lowball
  a pat Nine is typically a 1.8 to 1 favorite,
  a pat Ten is typically a slight favorite,
  a pat Jack is typically a slight underdog.
• In Deuce-to-Seven lowball, the same can be said about pat Tens, pat Jacks, and pat Queens, respectively.
• If Ace-to-Five lowball is played with the joker and the drawing hand holds the joker, a smooth one-card draw with the joker is typically a slight underdog against a pat Nine and a slight favorite against a pat Ten.

Pot-limit Omaha:

• A hand with two Aces is a favorite against all hands that do not have two Aces, except in a few rare cases. The worst case is A A 6 2 vs. J 10 9 8. In this match-up, the hand with the Aces wins about 49% of the time.
• A set is normally a favorite against a complete wrap on the flop.
• A flush draw is normally a favorite against a complete wrap on the flop.

Some interesting but not too useful mathematical facts about poker games:

• In holdem, with no betting after the flop, all pairs are a favorite over Ace-King offsuit, by at least 52% to 48%. All pairs except Deuces with neither card in the Ace-Kings suit are favored over Ace-King suited. If several players fold first, Ace-King suited is a favorite over most pairs. (The exceptions are Aces, Kings, and Jacks, and also Tens where one of the Tens is the same suit as the Ace-King.) Even Ace-King offsuit is now a favorite against small pairs. The reason for this is that players are more likely to play hands having an Ace or King than those containing smaller cards. Therefore, as players fold, the probability of an Ace or King coming on the board increases.
• The chances of having pocket Aces in holdem are 1 in 221. When everyone folds to you and you have the big blind, the probability of having a pair of Aces is approximately 1 in 134, since players are more likely to enter the pot if they have an Ace.
• Similarly, but more obviously, when everyone folds to the big blind in lowball, it is likely that the big blind will pick up a hand mainly consisting of low cards.
• When playing heads-up triple-draw lowball, it shouldn't be surprising that pat hands get beat on the third draw much more frequently than in single draw, since some useless cards have been discarded on the first two draws, and the opponent is drawing to better hands.
• Two players play a heads-up session for eight hours. Let's suppose they play 40 hands per hour for a total of 320 hands. If every hand has a 50% chance to be won by one player or the other, what is the average longest winning streak either of them will have?
  If n is the number of hands, the average longest streak is approximately logn even for small values of n. We should expect, on average, that one player will win eight hands in a row over the course of 320 hands. People tend to expect things to even out sooner than they do, so they are surprised by the streaks that come about. This leads them to believe that other things must be influencing the results: "He's just hot," "That's his lucky dealer," etc.