44 vs JTs vs AKo and binomial probability

44 vs JTs vs AKo
I recently heard about Amarillo's Slim hustle, where he'd offer you your choice of three Hold'em hands: 4 4, J T, or A K. He'd then choose one of the remaining two hands and deal out a flop, turn, river, with the best hand winning $100.

The key is the following edges:
Js Ts 53.65%
4c 4d 46.35%

4c 4d 53.94%
Ac Kh 46.06%

Ac Kh 57.23%
Js Ts 42.77%

I know that over cards versus under pairs are coin flips, with a slight edge usually going to the pairs. I was surprised to see that J T is a favorite over the 4 4 and thought it was just a matter of the cards being suited.

But 4 4 is still a favorite over (51.23% vs 48.77%) over A K, so it's not just the suitedness.

Is it just the added straight possibilities that make J T a slight favorite over 44 while A K a slight dog?

And with Slim's hustle, with the edges so small, how many times does he need to run this to really turn a real profit?

And finally, how did he figure out these slight edges without poker software?

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And with Slim's hustle, with the edges so small, how many times does he need to run this to really turn a real profit?

This is just binomial probability, which you can approximate pretty simply. Say he has a 54-46 edge, and he plays n hands. Then he "expects" to win 0.54n times (that's his Expected Value, or EV). Now, his standard deviation is √[n*p*(1-p)], where p is the probability he wins, so p = 0.54. That works out to be almost exactly (√n) / 2. Two thirds of the time, if he plays n hands, the number of times he wins will be within one standard deviation of his expected value, 95% of the time it will be within two standard deviations of his EV, and well over 99% of the time it will be within 3 standard deviations.

That may sound complicated, but it's easy to work out how likely he is to profit over, say, 400 hands. Then he would 'expect' to win 0.54*400 = 216 times. His standard deviation is (√400) / 2 = 10. So 2/3 of the time he will win between 206 and 226 times (add and subtract one standard deviation from his EV), for a sure profit, and 95% of the time he will win between 196 and 236 times. You could check a stats table to see how often he will end up with at least 200 wins precisely, but it will be around 85% the time I'd guess. The longer he plays, the more likely it becomes he ends up in profit.

转自2plus2.

呵呵，有点意思

呵呵，有意思。石头，剪子，布。

And finally, how did he figure out these slight edges without poker software?

这个的确值得考虑。硬算也能算出来，但我估计把这3个概率全都算出来，至少也得1整天，还得精神状态好不能算错。

我见过slim本人，看样子不像那种能踏实下来算这些东西的。我估计十有八九是他付钱给某大学生算得。