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OT: poker hypothetical

Posted By: Steve Mellen
Date: Wednesday, 3 February 2010, at 4:59 a.m.

In Response To: OT: poker hypothetical (Chuck Bower)

As a pure probability program the math seems easy. If you hold KK there are 6 ways the opponent can hold AA, 1 way he can hold KK, 8 ways he can hold AK. If you hold QQ there are 6 ways he can hold AA, 6 ways he can hold KK, 16 ways he can hold AK. Now, this is assuming that there are only three hands he will make this play with, like the problem says, but it ALSO assumes he will make this play 100% of the time when he holds one of those hands. If, for example, he would make this play 100% of the time with AA or KK, but only 50% of the time with AK, you need to adjust the likelihood of his holdings. Let's ignore this for now because we have no way to quantify these probabilities.

A lower pair beats a higher pair ~20% of the time, KK beats AK ~70% of the time, QQ beats AK ~57% of the time. Summing the probabilities in a back of the envelope sort of way, KK wins ~49% of the time, QQ wins ~41%. These are the odds if you deal out the rest of the hand from here.

But given that it's a deep stacks game, you don't have to play the hand through to the end. You can decide, for example, that you're going to call with your QQ and give up the hand if an ace or king hits on the flop. Since there are 48 cards left in the deck and 6 of them are aces or kings, the chance of getting none of them on the flop is 66%. Presumably you can raise him off the hand in those 66% of cases, so this adjusts your equity upwards in the 57% of cases where you're up against AK. I'm not sure we can calculate how much because it depends on exactly how many chips you're going to win or lose in these various non-all-in scenarios.

Likewise, with KK you can decide to give it up whenever an ace hits the flop (this will be embarrassing when your opponent also holds KK, but that's the way it goes). However, you gain less equity in this scenario than you gain in the scenario where you hold QQ, because an ace won't hit the flop as often as an ace/king will in the other scenario and thus you have fewer opportunities to save equity.

Like I said, the raw math is straightforward, but you can still make this problem as complicated as you like...

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