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| The Basics of the Mathematics of Poker |
Many times, questions about probability in poker games come up in discussion. “I went all-in with pocket kings before the flop, but somebody woke up with aces! What are the odds of that?” “What are the chances of winning for AK versus 88?” “My opponent flopped a set to beat my overpair. How about those bad beats?”
If you've ever wondered how to approach those questions, some basic poker math is needed.
Unknown Cards
One concept basic to all poker math is that of unknown cards. Although cards have been dealt to other players, we do not know what any of them are, so they do not affect the probability calculations we need to perform. If five players at a table receive AK, QJ, T9, 87, and 64 of hearts, and the flop has two hearts, how would each player calculate their chance of filling the flush? The answer is that nobody can assume less than nine outs, because they have no way of knowing what the other dealt hands were. (Pop quiz: who would win the hand if the thirteenth heart did come?)
Dealing Two Cards
In Texas holdem, each player receives two hole cards. There are 52 unknown cards when the dealer sends out our first card. (Remember, that even though other players got cards before us, we still don't know what they are.) Once we know our first card, there are 51 unknowns left in the deck. We have an equal probability of getting any one of them, so there are 52 x 51 = 2652 possible starting hands. Since the order we receive the cards does not affect the strength of the hand, we further divide that number by two to get the result of 1326 possible starting hands.
Since suits don't affect the strength of the hand (although having two of the same suit does), we can further eliminate duplicate starting hands. In this calculation, 7 - Clubs 4 - Diamonds is the same as 7 - Spades 4 = Hearts Now we reveal three basic hand types: pocket pairs (two cards of the same rank), suited cards (two cards of different ranks but the same suit), and unsuited cards (rank and suit both different). There are 13 possible pocket pairs, 78 suited combinations, and 78 unsuited combinations, for a total of 169 possible starting hands. This number is much more manageable than 1326.
For the suited combinations, the first card can be any of thirteen ranks, and the second is one of twelve. 13X12 = 156, and since order of delivery doesn't matter, we divide by two, revealing 78 suited rank combinations (AK, AQ...A2, KQ, KJ...K2, QJ, and so on). The same calculation applies for the unsuited cards.
Probability Versus Odds
Probability is a term used to express the chance that something will happen. Odds is the term used to express the chance that something will not happen. If I pick one card of 52, there is a 0.077 (7.7%) probability that it will be a seven (ignoring suits). The odds of it being a seven are 12-1 against.
Getting Pocket Aces
One common poker math question concerns the probability of getting dealt pocket aces. The answer is that there are six possible ways to get two aces out of 52 cards: A - Spades A - Hearts, A - Spades A - Diamonds, A - Spades A - Clubs, A - Diamonds A - Clubs. There are 1326 possible two card combinations, and six of them are pocket aces, so the probability of getting aces is 6/1326 = 1/221 = 0.0045, or less than one half of one per cent.
Getting Dominated
If you are holding a pocket pair, you should like to know the chance that someone holds a pair higher than yours. If you have KK, there are (50x49)/2=1225 possibilities for the pocket cards of any opponent. Six of those hands are pocket aces. The probability that any specific opponent has AA is 6/1225 = 0.49%. For 8 opponents at a nine-handed table, the probability that any one of them has AA is 1 - (1219/1225)^8 = 0.0385, or 3.85%.
If you have QQ, the probability that one opponent has AA or KK is 1- (1213/1225)^8 = 7.57%.
The study of poker mathematics is a fascinating subject if you are positively inclined towards math at all. There are a million probabilities that can be calculated, and studying odds away from the table can improve your play.
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Poker News Today.com © 2007 The Basics of the Mathematics of Poker
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