44 vs JTs vs AKo and binomial probability

dolphin · 发表于 2011-12-16 02:44:16

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.