KK vs AA, how often in 6 max game?

[Jocke_F]

Was just wondering what the odds of facing AA when you hold KK in a 6 handed game is, feels like its happening way more then it should on 6 max at party. Maybe I am way of but I have checked my pt logs and 1 in 15 of my KKs over the last 200 has faced AA is this about how often it should happen?

[TomCollins]

5 Opponents, odds none of the opponents has aces is odds each opponent doesn't have aces^5 = (25*49-6)/(25*49) ^ 5 = (.995)^5 = 97.6%. So the odds one (or more) has aces is 2.4%, or roughly 40-1 against it.

I'm not sure what you mean by "last 200". If you mean last 200 times that you had KK, someone had AA, thats a lot.

But anything that has this low frequency has pretty high variance. Tough luck.

[AaronBrown]

The answer is about right, but this is not precisely the way to compute it. The trouble is if you know one opponent does not have a pair of aces, it increases the probability that another one does.

Your opponents hold 10 cards between them out of the 50 cards you haven't seen. There are 10,272,278,170 ways to deal 10 cards out of 50. 4,076,350,421 of them have no aces, 4,406,865,320 have one ace, 1,565,596,890 have two aces, 214,098,720 have three aces and 9,366,819 have four aces.

If there are zero or one ace among your opponents, there are clearly no pairs. With 2 aces there's 1 chance in 9 of a pair, three aces make it 1 chance in 3, four aces make it 2 chances in 3 (this includes 1 chance in 21 that two players will have paired aces).

We have to multiply each of the chances above by the probability of facing 2, 3 or 4 aces; then add them up. The result is 2.45%.

With average luck, this will happen 4 or 5 times out of 200 chances. By the way, it doesn't matter what you hold, as long as neither card is an Ace. About 6% of the time it will happen 9 or more times out of 200. About 1% of the time it will happen 11 or more times out of 200. Only about 1 time in 700 will it happen 13 or more times out of 200, which is about 1 time in 15.

[AaronBrown]

One thing that might bias your perception is that if someone holds two aces, they're much more likely than average to stay in until showdown. Suppose that over the last 400 times you had KK, at least one opponent had AA 17 times. That's above the expected 10 times, but it will happen more than 1% of the time. Let's say that 200 times there was no showdown, so you didn't know what other players held, but of the 200 showdowns, 13 included ones with the pocket aces. You would see pocket aces 1 time in 15 you had KK, when you saw the hands.

[BruceZ]

Or you can just do

5*6/C(50,2) - C(5,2)/C(50,4)

=~ 2.44% or 39.9-to-1.

By the inclusion-exclusion principle.