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!