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Followup to Triptanes' question:
Can someone run the numbers for flush on flush (assuming suited pocket cards and a hand down to the river)? I know it's rare, but is it as rare as say, set on set?
I'm thinking of an occurence that sometimes happens in a lower-buyin cash game: a limped suited ace and a suited blind playing for a flush.
Thanks.
I have asked this before. I don’t know what the answer is but it seems like about 30% assuming a 10 handed ring game where see flop % > 50.
(x, y) = x!/(y! * (x-y)!)
Probability of dealing 2 suited hands in a 10 person game:
The 2 suited hands:
(13, 2) * (11, 2) / 2 = 2145
8 other hands:
(48, 16) * (16-1)!! = 4.57 * 10 ^ 18
Total with 2 suited:
2145 * 4.57 * 10 ^ 18 = 9.8 * 10 ^ 21
Total possible:
(52, 20) * (20 - 1)!! = 8.25 * 10 ^ 22
Probability
9.8 * 10 ^ 21 / 8.25 * 10 ^ 22 = 0.1188
There are 8 other hands, but we don't know what they are. We can assume that we're drawing cards for the board from the remaining 48 cards in the deck. If you knew someone had sooted cards in their hand it would change the probability, but it would also change the question.
Probability of a sooted board
3 soots on board (including 2 unsooted):
(9, 3) * (48-13, 2) = 49 980
4 soots on board (including 1 unsooted):
(9, 4) * (48-13, 1) = 4 410
5 soots on board:
(9, 5) = 126
Total suited boards:
49 980 + 4 410 + 126 = 54 516
Total boards:
(48, 5) = 1 712 304
Probability:
54 516 / 1 712 304 = 0.0318
Finally, the answer:
0.1188 * 0.0318 = 0.0038 or ~0.4%
Odds:
(1 - 0.0038) / 0.0038
263 to 1