Flopping an ace

Tony "TBC" writes in his latest blog post:

almost the entire table limps $2 and the lady on my right who just sat who i thinks gay and her partner just sat on my left and she makes it $12. shed been raising a lot the $15 min shed been there, so i wasnt worried about her. i had KK and made it $40 in my blind. everyone calls but the black guy who always calls me it seems cause hes got no respect for my game. (what a fish to call $40 preflop with his hand.) she folds of course. flop comes A57 and the pots $100+ and if im beat im beat, he will bet anyway maybe but to make sure i get his money if he has a hand like QQ i bet the last of it (about $110) and he calls and wins with AT offsuit.

i think pokerdogg would say if ur going to check/fold KK heads up anytime theres a ton of money in the pot and an ace comes u way too weak.

As many other people smarter than me have observed, no-limit hold'em is a game of implied odds, one in which capturing your opponent's entire stack needs to be your goal. Conversely, you need to avoid giving your opponent correct implied odds to make a better hand by being able to fold when he gets there, thus protecting your stack against your opponent trying to win it.

Let's simplify this problem to clarify whether Tony's opponent is indeed being a "fish" to call with an ace, using the apparently justified assumption that Tony will pay him off every time an ace flops, but that our man will be able to fold every time an ace does not flop. In other words, suppose our "fish" knows that Tony has exactly K-K.

The action isn't entirely clear, but let's guess that there were five limpers before the raise to $12, then our man in question called, then Tony reraised to $40. (Tony says "everyone calls but the black guy," but context suggests that this is a mistake and the word should have been "folds" rather than "calls.") So now he has to decide whether to call an additional $28 into a pot of about $74.

How often will an ace hit the flop if there are three left in the deck? We have four cards known (our A-x and Tony's K-K), leaving 48 unknown. There are 17,296 different flops possible from 48 unknown cards. Of those, 6625 will contain at least one ace. (To be precise, 6486 will have one ace, 138 will have two aces, and one will have all three remaining aces--if I've done the math right, which at this hour of the night is always a little dubious.) That's about 38% of the time.

That means that 62% of the time he loses the additional $28 he invests pre-flop. But 38% of the time he will win not only the $74 currently in the pot, but also the $110 that Tony has left in front of him. If they play this hand 100 times, he will lose 62 x $28 = $1736, but he will win 38 x $184 = $6992. As poker bets go, that's a great investment.

Of course things aren't really this simple. Sometimes the flop will have both an ace and a king, giving Tony a set, and our man will lose. Sometimes the flop will be A-x-x but a king will hit the turn or river after all the money is in. Sometimes one or the other of them will hit some weird runner-runner combination for a straight or flush. Sometimes his kicker will flop trips even without an ace. Etc. Also, he can't ever know that Tony has exactly K-K. Sometimes he'll actually be up against a bigger ace (though the probability of an ace hitting the flop then goes down drastically so it's easier for him to get away from it). On the other hand, I suspect Tony will do just about the same play with Q-Q and J-J, making the call with A-x even more profitable than if K-K were the only hand Tony played this way.

The point is that it is not at all clear that one is being a "fish" to call the reraise in this spot if one has good reason to believe that when an ace flops one will get one's opponent's entire stack most of the time.

I am not saying that one has to just give up with big pairs every time an ace flops. It's more nuanced than that, depending on how many people are in, what their pre-flop calling ranges are, how often they will represent having an ace when they don't, how often they will call a shove with lesser hands (e.g., the Q-Q Tony thought he might have), and so forth.

But the reality is that A-x hands are among the most common ones with which loose opponents--both good ones and bad ones--will call pre-flop. So while one need not necessarily surrender every time an ace shows up, it is also demonstrably bad practice to blindly charge ahead, ignore the potential danger, and shove every time. That pattern is what produces the favorable implied odds for an opponent to make a big call pre-flop.

Before concluding that somebody else is the fish in the game, it's good to check one's own mouth to see if there might be a hook there.

Addendum, August 25, 2011, 1:30 p.m.:

"Jasper6294" submitted a comment politely suggesting that my math was wrong. One of these years, I'll learn that if I write a heavily math-dependent post in the middle of the night, I should not hit "publish" until I've rechecked the number after a full night of sleep. Jasper is exactly correct, though it took me a while to figure out what I had done wrong. First I failed to remove the extra aces from my virtual deck before counting how many cards we had from which to deal the other cards on the flop. Second, in doing the one-ace flops, I double-counted, forgetting that it makes no difference what order the other cards come in, e.g., 9h-6c or 6c-9h.

I want to leave the evidence of my own occasional boneheadedness intact above, so I'm not editing the post as originally written. But here are the two math-dependent paragraphs rewritten correctly:

How often will an ace hit the flop if there are three left in the deck? We have four cards known (our A-x and Tony's K-K), leaving 48 unknown. There are 17,296 different flops possible from 48 unknown cards. Of those, 3106 will contain at least one ace. (To be precise, 2970 will have one ace, 135 will have two aces, and one will have all three remaining aces--if I've done the math right, which at this hour of the night is always a little dubious.) That's about 18% of the time.

That means that 82% of the time he loses the additional $28 he invests pre-flop. But 18% of the time he will win not only the $74 currently in the pot, but also the $110 that Tony has left in front of him. If they play this hand 100 times, he will lose 82 x $28 = $2296, but he will win 18 x $184 = $3312.

I left out the sentence about that being a "great" investment. The corrected numbers make it a good investment, but not what I would call a great one.

The larger point remains, though with a qualification that I should have added the first time around: One is not being a "fish" to call the reraise in this spot if one has good reason to believe that when an ace flops one will get one's opponent's entire stack most of the time and if the opponent's stack is large enough to make it worth the gamble. I understand this caveat deeply enough that I routinely check an opponent's stack size before calling a pre-flop raise with a speculative, tricky hand such as suited 6-7. That's because I know that it will be relative rare that I make a strong hand at the same time that he has enough of a hand that he will call off his stack. That means that the payoff has to be quite large when it happens to compensate for all the times that I miss and all the times that he gets away from a second-best hand.

In this case, as you can see from the revised math, Tony's stack was not huge, but was nevertheless large enough to make it worth an 18% chance of busting him. From a mathematical point of view, it was a perfectly justifiable call, given the assumptions stated above.