# How do you express the statistics of turn/river possibilities to make a hand?

This is more of a "understanding statistics" question than a poker question.

Faced with a question of, for example, "What are the odds of making a diamond royal flush by the river", the answer must encompass the players hole cards (5/52)*(4/51) and the cards coming on the flop turn and river. However, there are several different ways from a probability perspective that the royal could be hit. You could flop it, you could turn it (flop 2 + turn), you could flop 2, blank the turn but river it, or you could flop 1, turn the draw and river it.

So your formula has to account for the fairly easy flop it solution, plus the flop 2 (so 2 of 3 cards hit and 1 miss) plus turn, plus every other permutation...

How do you represent that, mathematically?

• Do you want to know how to calculate the odds from a mathematical point of view? (=> see links in Paparazzi's answer) Or do you want to know how they calculate those % you see on some televised poker shows or on some pokersites when all-in? Or do you want to know how to guesstimate those odds on the fly (in a hand)? May 17, 2016 at 10:20
• I want to understand the statistical method of determining odds when there are multiple permutations by which an outcome can be achieved. May 17, 2016 at 12:22
• Then you have that answer. That is the method used in the wiki link for odds of every hand. May 17, 2016 at 12:24
• Then Paparazzi's answer is what you want to look at (the combination part especially). May 17, 2016 at 12:24

Poker_probability

c(4,1) * c(5,5) * c(47,2) / c(52,7) = 0.0000323206 = 30939 : 1

4 flushes * exactly 5 cards * 47 remaining cards for the other 2

c is combination

It is combin in Excel

• Could you elaborate on what the terms mean? May 16, 2016 at 18:36
• c is combination May 16, 2016 at 18:49
• If you include a royal flush on the board (not using your holecards) this answer gives the odds of any royal flush. The diamond one would require multiplying the 30939 by 4 as you only want 1 of the 4 possible royal flushes (again including the royal flush on the board). May 17, 2016 at 10:07
• And to clear up any possible confussion, in the formula the "multiplying the 30939 by 4" means you remove the c(4,1) as the formula basically says "there are 4 possible royal flushes so multiply the chance of hitting a specific royal flush by 4 (= c(4,1) )". May 17, 2016 at 10:16
• @Paparazzi Ok, my bad. 30939 : 1 = 1 / 30940, so still a fraction but I was indeed not paying attention (thanks for pointing that out). So then multiply 30940 by 4 and you get 1 / 123760 which = 123759 : 1. The point however still being he asked for a the chance of a specific royal flush and you provided the chance of any royal flush. So you might want to address that bit in your answer. May 17, 2016 at 12:58