KK vs AA Why is My Math Wrong? (Inclusion-Exclusion and Overlapping Sets)
The math question is: What % of the time will KK flop at least 1 K & AA will not flop an ace?
My math is as follows:
You have to flop a K: 2(C,1)
Then you have a total of: 45 unseen cards that can complete the last 2 cards on the Flop: [52 - 2 As - 3 Ks, as 1 K is on the Flop] unseen cards - the 2 As that can't come on the Flop.
45(C,2)
So, it's [2(C,1)] * [45(C,2)] = 2 * 990 = 1980 possible Flops that bring at least one K & no As.
We have 48 unseen cards pre, so total possible Flops:
C(48,3) = 17,296
1980/17,296 = .11447 or 11.48% chance.
I've seen on at least 3 poker sites that it's in the low 20% arena, i.e., 20.17%, 21.67% & I think 22.06% are the numbers I've seen.
Or maybe what I've seen it quoted as is 20.17% 21.xx% & one other [that I can't remember but think is was 22.06%] on various sites.
What's up with my poor math skillz? 
2 Replies
Mike Caro had a section in SuperSystem very similar to this regarding a five card draw hustle about how often youll start with the two red jacks in your hand.
The math question is: What % of the time will KK flop at least 1 K & AA will not flop an ace?
My math is as follows:
You have to flop a K: 2(C,1)
Then you have a total of: 45 unseen cards that can complete the last 2 cards on the Flop: [52 - 2 As - 3 Ks, as 1 K is on the Flop] unseen cards - the 2 As that can't come on the Flop.
45(C,2)
So, it's [2(C,1)] * [45(C,2)] = 2 * 990 = 1980 possible Flops that bring at least one K & no As.
We have 48 unseen cards pre, so total possible Flops:
C(48,3) = 17,296
The probability of one king and no ace is 2/48 x 44/47 x 43/46 x 3. Two kings and no ace is 2/48 x 1/47 x 44/46 x3. The total probability of 11616/103776 when multiplied by the total number of different flops (17296) is 1936.
