Hand Frequency Probability Cumulative Odds Mathematical expression of frequency

Royal flush 4 0.000154% 0.000154% 649,739 : 1

{4 \choose 1}

Straight flush

(excluding royal flush)

36 0.00139% 0.00154% 72,192.33 : 1

{10 \choose 1}{4 \choose 1} - {4 \choose 1}

Four of a kind 624 0.0240% 0.0256% 4,164 : 1

{13 \choose 1}{4 \choose 4}{48 \choose 1}

Full house 3,744 0.144% 0.170% 693.2 : 1

{13 \choose 1}{4 \choose 3}{12 \choose 1}{4 \choose 2}

Flush 5,108 0.197% 0.367% 507.8 : 1

{13 \choose 5}{4 \choose 1} - {10 \choose 1}{4 \choose 1}

Straight 10,200 0.392% 0.76% 253.8 : 1

{10 \choose 1}{4 \choose 1}^5 - {10 \choose 1}{4 \choose 1}

Three of a kind 54,912 2.11% 2.87% 46.3 : 1

{13 \choose 1}{4 \choose 3}{12 \choose 2}{4 \choose 1}^2

Two pair 123,552 4.75% 7.62% 20.03 : 1

{13 \choose 2}{4 \choose 2}^2{11 \choose 1}{4 \choose 1}

One pair 1,098,240 42.3% 49.9% 1.37 : 1

{13 \choose 1}{4 \choose 2}{12 \choose 3}{4 \choose 1}^3

No pair / High card 1,302,540 50.1% 100% 0.995 : 1

\left[{13 \choose 5} - 10\right]\left[{4 \choose 1}^5 - 4\right]

Total 2,598,960 100% 100% 0 : 1

{52 \choose 5}

If this is confusing to anyone, a graphical version can be viewed here:

Poker probability - Wikipedia, the free encyclopedia

Royal flush 4 0.000154% 0.000154% 649,739 : 1

{4 \choose 1}

Straight flush

(excluding royal flush)

36 0.00139% 0.00154% 72,192.33 : 1

{10 \choose 1}{4 \choose 1} - {4 \choose 1}

Four of a kind 624 0.0240% 0.0256% 4,164 : 1

{13 \choose 1}{4 \choose 4}{48 \choose 1}

Full house 3,744 0.144% 0.170% 693.2 : 1

{13 \choose 1}{4 \choose 3}{12 \choose 1}{4 \choose 2}

Flush 5,108 0.197% 0.367% 507.8 : 1

{13 \choose 5}{4 \choose 1} - {10 \choose 1}{4 \choose 1}

Straight 10,200 0.392% 0.76% 253.8 : 1

{10 \choose 1}{4 \choose 1}^5 - {10 \choose 1}{4 \choose 1}

Three of a kind 54,912 2.11% 2.87% 46.3 : 1

{13 \choose 1}{4 \choose 3}{12 \choose 2}{4 \choose 1}^2

Two pair 123,552 4.75% 7.62% 20.03 : 1

{13 \choose 2}{4 \choose 2}^2{11 \choose 1}{4 \choose 1}

One pair 1,098,240 42.3% 49.9% 1.37 : 1

{13 \choose 1}{4 \choose 2}{12 \choose 3}{4 \choose 1}^3

No pair / High card 1,302,540 50.1% 100% 0.995 : 1

\left[{13 \choose 5} - 10\right]\left[{4 \choose 1}^5 - 4\right]

Total 2,598,960 100% 100% 0 : 1

{52 \choose 5}

If this is confusing to anyone, a graphical version can be viewed here:

Poker probability - Wikipedia, the free encyclopedia

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