<Table> <Tr> <Th> Hand </Th> <Th> Distinct hands </Th> <Th> Frequency </Th> <Th> Probability </Th> <Th> Cumulative probability </Th> <Th> Odds </Th> <Th> Mathematical expression of absolute frequency </Th> </Tr> <Tr> <Td> Royal flush <P> </P> </Td> <Td> </Td> <Td> </Td> <Td> 0.000154 % </Td> <Td> 0.000154 % </Td> <Td> 649,739 : 1 </Td> <Td> ( 4 1 ) ( \ displaystyle ( 4 \ choose 1 ) ) </Td> </Tr> <Tr> <Td> Straight flush ( excluding royal flush ) <P> </P> </Td> <Td> 9 </Td> <Td> 36 </Td> <Td> 0.00139 % </Td> <Td> 0.0015 % </Td> <Td> 72,192 : 1 </Td> <Td> ( 10 1 ) ( 4 1 ) − ( 4 1 ) ( \ displaystyle ( 10 \ choose 1 ) ( 4 \ choose 1 ) - ( 4 \ choose 1 ) ) </Td> </Tr> <Tr> <Td> Four of a kind <P> </P> </Td> <Td> 156 </Td> <Td> 624 </Td> <Td> 0.0240 % </Td> <Td> 0.0256 % </Td> <Td> 4,164 : 1 </Td> <Td> ( 13 1 ) ( 12 1 ) ( 4 1 ) ( \ displaystyle ( 13 \ choose 1 ) ( 12 \ choose 1 ) ( 4 \ choose 1 ) ) </Td> </Tr> <Tr> <Td> Full house <P> </P> </Td> <Td> 156 </Td> <Td> 3,744 </Td> <Td> 0.1441 % </Td> <Td> 0.17 % </Td> <Td> 693 : 1 </Td> <Td> ( 13 1 ) ( 4 3 ) ( 12 1 ) ( 4 2 ) ( \ displaystyle ( 13 \ choose 1 ) ( 4 \ choose 3 ) ( 12 \ choose 1 ) ( 4 \ choose 2 ) ) </Td> </Tr> <Tr> <Td> Flush ( excluding royal flush and straight flush ) <P> </P> </Td> <Td> 1,277 </Td> <Td> 5,108 </Td> <Td> 0.1965 % </Td> <Td> 0.367 % </Td> <Td> 508 : 1 </Td> <Td> ( 13 5 ) ( 4 1 ) − ( 10 1 ) ( 4 1 ) ( \ displaystyle ( 13 \ choose 5 ) ( 4 \ choose 1 ) - ( 10 \ choose 1 ) ( 4 \ choose 1 ) ) </Td> </Tr> <Tr> <Td> Straight ( excluding royal flush and straight flush ) <P> </P> </Td> <Td> 10 </Td> <Td> 10,200 </Td> <Td> 0.3925 % </Td> <Td> 0.76 % </Td> <Td> 254 : 1 </Td> <Td> ( 10 1 ) ( 4 1 ) 5 − ( 10 1 ) ( 4 1 ) ( \ displaystyle ( 10 \ choose 1 ) ( 4 \ choose 1 ) ^ ( 5 ) - ( 10 \ choose 1 ) ( 4 \ choose 1 ) ) </Td> </Tr> <Tr> <Td> Three of a kind <P> </P> </Td> <Td> 858 </Td> <Td> 54,912 </Td> <Td> 2.1128 % </Td> <Td> 2.87 % </Td> <Td> 46.3 : 1 </Td> <Td> ( 13 1 ) ( 4 3 ) ( 12 2 ) ( 4 1 ) 2 ( \ displaystyle ( 13 \ choose 1 ) ( 4 \ choose 3 ) ( 12 \ choose 2 ) ( 4 \ choose 1 ) ^ ( 2 ) ) </Td> </Tr> <Tr> <Td> Two pair <P> </P> </Td> <Td> 858 </Td> <Td> 123,552 </Td> <Td> 4.7539 % </Td> <Td> 7.62 % </Td> <Td> 20.0 : 1 </Td> <Td> ( 13 2 ) ( 4 2 ) 2 ( 11 1 ) ( 4 1 ) ( \ displaystyle ( 13 \ choose 2 ) ( 4 \ choose 2 ) ^ ( 2 ) ( 11 \ choose 1 ) ( 4 \ choose 1 ) ) </Td> </Tr> <Tr> <Td> One pair <P> </P> </Td> <Td> 2,860 </Td> <Td> 1,098,240 </Td> <Td> 42.2569 % </Td> <Td> 49.9 % </Td> <Td> 1.37 : 1 </Td> <Td> ( 13 1 ) ( 4 2 ) ( 12 3 ) ( 4 1 ) 3 ( \ displaystyle ( 13 \ choose 1 ) ( 4 \ choose 2 ) ( 12 \ choose 3 ) ( 4 \ choose 1 ) ^ ( 3 ) ) </Td> </Tr> <Tr> <Td> No pair / High card <P> </P> </Td> <Td> 1,277 </Td> <Td> 1,302,540 </Td> <Td> 50.1177 % </Td> <Td> 100 % </Td> <Td> 0.995 : 1 </Td> <Td> ( ( 13 5 ) − 10 ) ( ( 4 1 ) 5 − 4 ) ( \ displaystyle \ left ( ( 13 \ choose 5 ) - 10 \ right ) \ left ( ( 4 \ choose 1 ) ^ ( 5 ) - 4 \ right ) ) </Td> </Tr> <Tr> <Th> Total </Th> <Th> 7,462 </Th> <Th> 2,598,960 </Th> <Th> 100 % </Th> <Th> -- - </Th> <Th> 0 : 1 </Th> <Th> ( 52 5 ) ( \ displaystyle
<Table> <Tr> <Th> Hand </Th> <Th> Distinct hands </Th> <Th> Frequency </Th> <Th> Probability </Th> <Th> Cumulative probability </Th> <Th> Odds </Th> <Th> Mathematical expression of absolute frequency </Th> </Tr> <Tr> <Td> Royal flush <P> </P> </Td> <Td> </Td> <Td> </Td> <Td> 0.000154 % </Td> <Td> 0.000154 % </Td> <Td> 649,739 : 1 </Td> <Td> ( 4 1 ) ( \ displaystyle ( 4 \ choose 1 ) ) </Td> </Tr> <Tr> <Td> Straight flush ( excluding royal flush ) <P> </P> </Td> <Td> 9 </Td> <Td> 36 </Td> <Td> 0.00139 % </Td> <Td> 0.0015 % </Td> <Td> 72,192 : 1 </Td> <Td> ( 10 1 ) ( 4 1 ) − ( 4 1 ) ( \ displaystyle ( 10 \ choose 1 ) ( 4 \ choose 1 ) - ( 4 \ choose 1 ) ) </Td> </Tr> <Tr> <Td> Four of a kind <P> </P> </Td> <Td> 156 </Td> <Td> 624 </Td> <Td> 0.0240 % </Td> <Td> 0.0256 % </Td> <Td> 4,164 : 1 </Td> <Td> ( 13 1 ) ( 12 1 ) ( 4 1 ) ( \ displaystyle ( 13 \ choose 1 ) ( 12 \ choose 1 ) ( 4 \ choose 1 ) ) </Td> </Tr> <Tr> <Td> Full house <P> </P> </Td> <Td> 156 </Td> <Td> 3,744 </Td> <Td> 0.1441 % </Td> <Td> 0.17 % </Td> <Td> 693 : 1 </Td> <Td> ( 13 1 ) ( 4 3 ) ( 12 1 ) ( 4 2 ) ( \ displaystyle ( 13 \ choose 1 ) ( 4 \ choose 3 ) ( 12 \ choose 1 ) ( 4 \ choose 2 ) ) </Td> </Tr> <Tr> <Td> Flush ( excluding royal flush and straight flush ) <P> </P> </Td> <Td> 1,277 </Td> <Td> 5,108 </Td> <Td> 0.1965 % </Td> <Td> 0.367 % </Td> <Td> 508 : 1 </Td> <Td> ( 13 5 ) ( 4 1 ) − ( 10 1 ) ( 4 1 ) ( \ displaystyle ( 13 \ choose 5 ) ( 4 \ choose 1 ) - ( 10 \ choose 1 ) ( 4 \ choose 1 ) ) </Td> </Tr> <Tr> <Td> Straight ( excluding royal flush and straight flush ) <P> </P> </Td> <Td> 10 </Td> <Td> 10,200 </Td> <Td> 0.3925 % </Td> <Td> 0.76 % </Td> <Td> 254 : 1 </Td> <Td> ( 10 1 ) ( 4 1 ) 5 − ( 10 1 ) ( 4 1 ) ( \ displaystyle ( 10 \ choose 1 ) ( 4 \ choose 1 ) ^ ( 5 ) - ( 10 \ choose 1 ) ( 4 \ choose 1 ) ) </Td> </Tr> <Tr> <Td> Three of a kind <P> </P> </Td> <Td> 858 </Td> <Td> 54,912 </Td> <Td> 2.1128 % </Td> <Td> 2.87 % </Td> <Td> 46.3 : 1 </Td> <Td> ( 13 1 ) ( 4 3 ) ( 12 2 ) ( 4 1 ) 2 ( \ displaystyle ( 13 \ choose 1 ) ( 4 \ choose 3 ) ( 12 \ choose 2 ) ( 4 \ choose 1 ) ^ ( 2 ) ) </Td> </Tr> <Tr> <Td> Two pair <P> </P> </Td> <Td> 858 </Td> <Td> 123,552 </Td> <Td> 4.7539 % </Td> <Td> 7.62 % </Td> <Td> 20.0 : 1 </Td> <Td> ( 13 2 ) ( 4 2 ) 2 ( 11 1 ) ( 4 1 ) ( \ displaystyle ( 13 \ choose 2 ) ( 4 \ choose 2 ) ^ ( 2 ) ( 11 \ choose 1 ) ( 4 \ choose 1 ) ) </Td> </Tr> <Tr> <Td> One pair <P> </P> </Td> <Td> 2,860 </Td> <Td> 1,098,240 </Td> <Td> 42.2569 % </Td> <Td> 49.9 % </Td> <Td> 1.37 : 1 </Td> <Td> ( 13 1 ) ( 4 2 ) ( 12 3 ) ( 4 1 ) 3 ( \ displaystyle ( 13 \ choose 1 ) ( 4 \ choose 2 ) ( 12 \ choose 3 ) ( 4 \ choose 1 ) ^ ( 3 ) ) </Td> </Tr> <Tr> <Td> No pair / High card <P> </P> </Td> <Td> 1,277 </Td> <Td> 1,302,540 </Td> <Td> 50.1177 % </Td> <Td> 100 % </Td> <Td> 0.995 : 1 </Td> <Td> ( ( 13 5 ) − 10 ) ( ( 4 1 ) 5 − 4 ) ( \ displaystyle \ left ( ( 13 \ choose 5 ) - 10 \ right ) \ left ( ( 4 \ choose 1 ) ^ ( 5 ) - 4 \ right ) ) </Td> </Tr> <Tr> <Th> Total </Th> <Th> 7,462 </Th> <Th> 2,598,960 </Th> <Th> 100 % </Th> <Th> -- - </Th> <Th> 0 : 1 </Th> <Th> ( 52 5 ) ( \ displaystyle
what are the odds of getting a royal flush in poker
what are the odds of getting a royal flush in poker
649,739 : 1
649,739 : 1