Poker Probabilities

Poker probabilities involve the likelihood of various hand combinations occurring in the game of poker.

Calculating Probabilities

The probability of being dealt a specific hand is calculated by dividing the number of ways that hand can be formed by the total number of 5-card combinations possible from a 52-card deck. The total number of 5-card combinations is $(52C5)=2,598,960$.

$$n C r = n ! ( n − r ) ! r !$$

1. Royal Flush: The highest possible hand in poker, consisting of the Ace, King, Queen, Jack, and 10 of the same suit. The probability of this hand is $4/2,598,960≈0.000154$ because there are only four ways to form a Royal Flush, one for each suit.

2. Straight Flush: Any five consecutive cards of the same suit that are not a Royal Flush. There are 36 possible straight flushes (including the four Royal Flushes). The probability is $36/2,598,960≈0.00139$.

3. Four of a Kind: Four cards of the same rank and one card of a different rank. The probability is $624/2,598,960≈0.0240$.

4. Full House: Three cards of one rank and two cards of another rank. The probability of being dealt a full house is $3,7442,598,960≈0.1441$.

5. Flush: Any five cards of the same suit, not in sequence. The probability of a flush (excluding straight flushes) is $5,108/2,598,960≈0.1965$.

6. Straight: Five consecutive cards of different suits. The probability is $10,2002,598,960≈0.3925$.

7. Three of a Kind: Three cards of the same rank and two other cards of different ranks. The probability is $54,912/2,598,960≈2.1128$.

8. Two Pair: Two cards of one rank, two cards of another rank, and one other card. The probability is $123,552/2,598,960≈4.7539$.

9. One Pair: Two cards of one rank and three other cards. This is a very common hand with a probability of $1,098,240/2,598,960≈42.2569$.

10. High Card: Any hand that does not qualify as any of the above types. The probability is $1,302,540/2,598,960≈50.1177$.

Probability Explanations

Straight Flushes

Using straight flushes as an example, a straight flush is any sequence of five consecutive cards that are all of the same suit.

There are 10 different starting points for a sequence of five consecutive cards in any given suit. These sequences start from:

• Ace (considered high in this context)

• 2

• 3

• 4

• 5

• 6

• 7

• 8

• 9

• 10

Each sequence ends at a card that is four ranks higher than the starting card. For example, a sequence starting from 2 would end at 6 (2, 3, 4, 5, 6).

Count Across All Suits: Since each sequence can occur in any of the four suits, you multiply the 10 starting points by the 4 suits:

$$10 sequences×4 suits = 40 straight flushes$$

The initial calculation of 40 seems to cover all straight flushes, but here's where it's crucial to differentiate between straight flushes and Royal Flushes:

Royal Flush is a special kind of straight flush and is typically counted separately in poker statistics. It consists of the Ace-high straight flush (10, J, Q, K, Ace). There are 4 Royal Flushes, one in each suit.

If we exclude these 4 Royal Flushes from our count, we adjust our total to:

$$40 straight flushes - 4 royal flushes = 36 straight flushes$$

This calculation leaves us with 36 possible straight flushes (excluding Royal Flushes) across all suits, giving players these specific hand possibilities in a game of poker.