# Poker Math & Probability

A lot of games use mathematical logic, and poker is no exception. Along with psychology, math is a key factor in mastering this game. This article deals with the basics of probability and how it is applied in poker, including the probability of being dealt certain hands and how often these hands are likely to win.

Probability is the mathematical branch that looks at the likelihood that a certain outcome, among all the other possible outcomes, will occur. A very basic example is a coin toss which has only two possible outcomes, heads or tails. Getting a head has a probability of 50%, since getting a head is one out of two possible outcomes. The same is true with getting a tail, which obviously also has a 50% probability.

Things get slightly more complicated with poker as it uses a deck of 52 cards, with each card designated with one of four suits (spades, clubs, hearts and diamonds) and one of 13 ranks (Ace, numbers two to ten, Jack, Queen, King). Thus, the probability of getting any spade as your first card is 25% (i.e., one in four) while the probability of getting an Ace as your first card is 7.7% (i.e., one in 13 or four in 52). The probability of getting an Ace of spade is even lower at 1.9% (or, one in 52). Things are further complicated by the fact that, unlike coins, cards are said to have "memory", i.e., the makeup of the deck, and thus the probability, changes with every dealt card. To illustrate, once an Ace has been dealt, the probability of receiving another ace goes down to 5.9% from 7.7% as there are now only three Aces in the deck of 51 cards.

Let us now look at the poker probability of being dealt a particular hand, such as a pair of Aces. This is arrived at with a simple multiplication equation using the probability of each card:

(4/52) x (3/51) = (12/2652) = (1/221) or about 0.45%

From a practical view, this means that if a player is in a game dealing 30 hands per hour, he should expect to receive a pair of Aces only once every 7.5 hours (or after about 225 hands have been dealt).

As a poker player measures his hand against opponents, it is worthwhile to study the poker probability of most pre-flop situations:

YOU HOLD

YOUR OPPONENT HOLDS

YOUR PROBABILITY OF WINNING

High Pair

Two Low Cards

83.00%

High Pair

Low Pair

82.00%

Middle Pair

One High Card, one Low Card

71.00%

Two High Cards

Two Low Cards

63.00%

Two High Cards

Low Pair

55.00%