# How to calculate probability of flopping a set (with pocket pairs)?

Building a poker program and would like to make sure I have my calculation correct. If a player is holding pocket pair, how does one calculate the probability of flopping a set (one card out of the three will match the player's pair). Ex: Let's say I'm holding pocket Aces.

The way I am attempting to calculate this is by using the binomial formula and finding all 3-card combinations that include exactly one of the two remaining aces and divide that by all possible 3-card combinations (50 Choose 3 = 19,600)

• So given the Aces I'm already holding, I simply do 2 Choose 1 and calculate the number of combos I can have with the remaining Aces in the deck.
• Next, for the remaining two cards on the flop that cannot repeat, I do 12 Choose 2
• Then for each of these two cards there's 4 different possible suits so I do 4 Choose 1 = 4.
• Multiplying all of these together I get: 2 * 66 * (4 ^ 2) = 2,112

Finally, I get 2,112 / 19,600 = 10.78%

A few places online state that the chances of flopping a set with pocket pairs is around 11.5 - 11.8%. However I cannot find an in-depth explanation of the calculation that would help me modify the calculation I will use in my program.

Can someone please explain to me where I'm going wrong in my calculation? And if there's a simpler way to calculate this probability please explain? Thanks!

total combinations = 19,600

assuming you have red 2s

flops with 2s but no 2c [2s, 48, 47] = 48c2 = 1128

flops with 2c but no 2s [2c, 48, 47] = 48c2 = 1128

flops with 22x [2c, 2s, 48] = 48

2256/19600 = 11.5102%

combined = 11.755102%

I also saw it represented as the inverse of the probability that there will not be a 2 on the flop

48/50 = .96

47/49 = .95918

46/48 = .95833

= 103776/117600 = .88244897

inverse is .11755 or 11.75%

• In your first example, 48c2 should be 1128, which in the end would be 11.5% to flop a set. My problem was that I was calculating the probability of flopping ONLY a set, but your method makes more sense, is more simple, and explains where everyone got the 11.5%. Just what I was looking for, thanks! Dec 12, 2017 at 20:50
• good catch. yeah that makes sense. I was wondering why it was off as the two should give the same result Dec 12, 2017 at 20:52

jfviray, Your calculations are correct for flopping set exactly. 11.8% is the probability of flopping set or better which includes quads and full house too.

The above calculation by Kenny shows that