A Little Math Trick

You are in a tournament and have made the final table; nine players are left all with roughly the same amount of chips. One of your opponents, Jimmy, is a tight player you feel you have a great read on.

While sitting in middle position you are dealt QQ and you open the betting with a wager three times the big blind. Jimmy, on your left, pushes all in and everyone else folds. From your experience you are certain that Jimmy is holding either AA, KK or AK. If you call and he has rockets or cowboys, you are probably looking at a ninth place finish. If you call and he has AK, you are a slight favourite to become the chip leader- an upside that would make you take the gamble if you thought he was likely to have AK. So what do you do?

On the face of it, the answer seems to be "fold". If Jimmy has either AA, KK or AK, then that means there is a 2 out of 3 chance he has AA or KK, right?

Well, no. Actually it is more likely he has AK.

Let's do the math: There are 6 possible combinations of cards out of the deck that give him AA:

Ac Ah, Ad As, As Ah, As Ac, Ah Ad, Ad Ac

There are 6 possible combinations of cards in the deck that give KK:

Kc Kh, Kd Ks, Ks Kh, Ks Kc, Kh Kd, Kd Kc

However there are 16 combinations of cards that give Jimmy AK:

Ac Kc, Ac Kd, Ac Ks, Ac Kh

Ad Kc, Ad Kd, Ad Ks, Ad Kh

As Kc, As Kd, As Ks, As Kh

Ah Kc, Ah Kd, Ah Ks, Ah Kh

Of 28 different combinations of cards that give Jimmy AA, KK or AK 16 of them make the AK and only 12 make an over pair to your QQ. There is a 57% chance you are ahead of Jimmy right now.

Whether or not you decide it is a worthwile call, the point is to remember when calculating your chances against the various hands in your opponents range that the odds of being dealt any two specific cards of different ranks is much greater than the odds of being dealt a specific pair.