Poker Probability

Hand Frequency Approx. Probability Approx. Odds Mathematical Expression
Royal flush

10 of spades Jack of spades Queen of spades King of spades Ace of spades

4 0.000154% 649,739 : 1 {4 \choose 1}
Straight flush (excluding royal flush)

4 of hearts

36 0.00139% 72,192.33 : 1 {10 \choose 1}{4 \choose 1} - {4\choose 1}
Four of a kind

9 of hearts

624 0.0240% 4,164 : 1 {13 \choose 1}{12 \choose 1}{4 \choose 1}
Full house

8 of hearts

3,744 0.144% 693.2 : 1 {13 \choose 1}{4 \choose 3}{12 \choose 1}{4 \choose 2}
Flush (excluding royal flush and straight flush)

10 of clubs

5,108 0.197% 507.8 : 1 {13 \choose 5}{4 \choose 1} - {10 \choose 1}{4 \choose 1}
Straight (excluding royal flush and straight flush)

7 of clubs

10,200 0.392% 253.8 : 1 {10 \choose 1}{4 \choose 1}^5 - {10 \choose 1}{4 \choose 1}
Three of a kind

Queen of hearts Queen of clubs Queen of diamonds 5 of spades Ace of diamonds

54,912 2.11% 46.3 : 1 {13 \choose 1}{4 \choose 3}{12 \choose 2}{4 \choose 1}^2
Two pair

3 of hearts

123,552 4.75% 20.03 : 1 {13 \choose 2}{4 \choose 2}^2{11 \choose 1}{4 \choose 1}
One pair

5 of hearts

1,098,240 42.3% 1.36 : 1 {13 \choose 1}{4 \choose 2}{12 \choose 3}{4 \choose 1}^3
No pair / High card

2 of diamonds

1,302,540 50.1% .995 : 1 \left[{13 \choose 5} - 10\right]\left[{4 \choose 1}^5 - 4\right]
Total 2,598,960 100% 1 : 1 {52 \choose 5}