<Li> The values given for Probability, Cumulative probability, and Odds are rounded off for simplicity; the Distinct hands and Frequency values are exact . </Li> <P> The nCr function on most scientific calculators can be used to calculate hand frequencies; entering nCr with 52 and 5, for example, yields (52 5) = 2, 598, 960 (\ displaystyle (\ begin (matrix) (52 \ choose 5) = 2,598,960 \ end (matrix))) as above . </P> <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 (52 \ choose 5)) </Th> </Tr> </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>

What are the odds of getting a royal flush in poker