Poker Probability

Hand	Frequency	Approx. Probability	Approx. Odds	Mathematical Expression
Royal flush	4	0.000154%	649,739 : 1	${4 \choose 1}$
Straight flush (excluding royal flush)	36	0.00139%	72,192.33 : 1	${10 \choose 1}{4 \choose 1} - {4\choose 1}$
Four of a kind	624	0.0240%	4,164 : 1	${13 \choose 1}{12 \choose 1}{4 \choose 1}$
Full house	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)	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)	10,200	0.392%	253.8 : 1	${10 \choose 1}{4 \choose 1}^5 - {10 \choose 1}{4 \choose 1}$
Three of a kind	54,912	2.11%	46.3 : 1	${13 \choose 1}{4 \choose 3}{12 \choose 2}{4 \choose 1}^2$
Two pair	123,552	4.75%	20.03 : 1	${13 \choose 2}{4 \choose 2}^2{11 \choose 1}{4 \choose 1}$
One pair	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	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	${52 \choose 5}$