How do I calculate the odds of making a four of a kind by the river given I have a pair?

Given: My hand is any pair

How do I calculate the probability of making a four of a kind by the (flop, turn, &) river?

• +1 nice question. Keep them coming if you have any more :) Oct 21 '14 at 18:49

Without taking into account the fact that the very act of seeing the flop with one or several other player(s) influence the distribution of the flop (*), here's one way how you could compute these odds:

• you have C(50,3) possible flops: that is 19 600 flops
• out of these there are 48 cases where you'll improve directly to quads, so the probability to flop quads is 48 / C(50,3): 0.245%
• out of these C(50,3) possible flops there are 48*47 cases where you'll improve to a set or a full house, so the probability to flop a set or a full house is 48*47 / C(50,3): 11.51%
• if you have not flopped a set (nor a full house), your probability to make quads by the turn is zero.
• if you have flopped a set (or a full house), your probability to make quads by the turn is 1/47: 2.13%
• if you have flopped a set (or a full house), your probability to make quads by the river if you didn't make quads by the turn is 1/46: 2.17%
• if you have not flopped a set (nor a full house), your probability to make quads by the river is: 2/47 * 1/46: 0.08%

I may have miscalculated something here or there but this answer should give you the basic idea.

• (*) when you go to the flop your opponents is, for example, much more likely to hold AA than 72o. Studies over billions of online hands have shown that the distribution of flop cards is skewed due to this kind of "card removal effect" or whatever it is called. Oct 20 '14 at 2:57
• For the sake of completeness: there's 10.77% to flop a set, 0.74% to flop a full house and 0.245% to flop quads, which adds up to 11.75%. This 11.75% corresponds to the often mentioned "7.5:1" (1 / 8.5 is "7.5 to 1") odds to improve to a set or better at flop when you hold a pocket pair. Oct 20 '14 at 3:03
• This is off by a decimal point 48 / C(50,3) = 2.45% Dec 7 '16 at 16:47

Looking at just combination and only matching your pocket pair
Quads on the board is possible

flop
combin(2,2) * combin(48,1) / combin(50,3) = 0.245%

turn
combin(2,2) * combin(48,2) / combin(50,4) = 0.490%

river
combin(2,2) * combin(48,3) / combin(50,5) = 0.816%

It surprises me the chances double going from the flop to the turn but that is what these numbers say. You do double the number of free cards.

You are typically playing pocket pair for the set and quads is gravy (sometimes gravy you need).