2024

# Poker probability

The game of poker is quite popular with players today. Mathematics and the probability theory can be applied even to poker. In poker the probability of every type of five card hand can be calculated. This theory is particularly known as the poker probability.

## Frequency of the five card hands

Every hand in poker has an absolute frequency when all the combination of the five cards have been randomly drawn from a complete deck of 52 cards without any replacement. The wild cards are not taken into consideration. For example, the royal flush can be formed in four different ways, one for every suit.

## The derivation of the five card hands

The derivation of the five card hands in poker is as explained below:

• Straight flush

Every straight flush is determined by its highest ranking card. These cards can be from 5 to A and these again can be in four suits. Therefore, the total number of the straight flushes would be:

• Royal straight flush

A royal straight flush is the subset of all the straight flushes where the highest card is the ace. The total number of royal straight flush is:

• Four of a kind

Four of a kind can be any one of thirteen ranks and will have all four of the suits for the same rank.

• Full house

The full house will have a triple and a pair. The total number of such a hand is:

• Flush

A flush will have any five from the thirteen ranks but from only one suit:

• Straight

A straight will have any one of the ten sequences of five cards consecutively. These cards can be from any suit. The total number of flushes would then be:

• Three of a kind

Three of a kind can be any three cards of the same rank from any of the three suits. The total number of such a hand is:

• Two pairs

The pairs could have any two from the thirteen ranks and can be from any two of the four different suits. The total number of such a hand is:

• Pair

The pair can be one from the thirteen ranks and from any two of the total four suits. The total number of hands with pairs can be:

• No pair

No pair hand means a hand which contains five cards from the total thirteen ranks, other than the straights and the cards could be from any of the suits except for the flushes. The total number of hands with no-pairs is:

This is how the probability of the number of hands showing up in poker can be calculated. The poker probability had been developed in order to understand the frequency with which each of these types of hands can show up during a game. For example, studying the probability it is possible to say that the chance of a no pair hand showing up during the game is far higher than the chance of a three of a kind hand showing up.

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