more thoughts on the 3-bet - getting the math right (I hope).

I've been thinking about the math behind my light 3-betting, mentioned in my earlier post. I have since realized that my original math was incomplete.

In order to calculate the true equity of a play like this, I need to add in the times when I don't successfully graze the flop, and have to fold, as I did last night with my 6s7s. (thinking more about it, I should have indeed just folded on the dual broadway board and not taken my continuation bet stab).

So the total equity of the play must account for the following steps. I've included my very first guess of the percentages of each occurring -

villain folds flop to your 3 bet. (50%)
villain 4-bet shoves, you fold. (5% of the time he has JJ+/AK+ and some bluffs)
villain calls (45%)
--- you whiff flop completely and give up. (70% of the time he calls, you whiff or hit a too-weak hand to continue)
--- you hit the flop somehow and get it all in (30%),
-------- he folds (67%).
-------- he calls. (33%)

Your percentage of winning once he calls the all-in varies greatly with how hard you hit the flop, but let's assume that when he calls, he is currently ahead and you have to hit around 6 outs to win, so you win like 25% of the time and lose 75%. This seems reasonable to me - your opponent has already been established as a decent player who isn't going to call an all-in most of the time when way behind. We'll skip the few times (5%-ish) that you end up ahead by flopping 2 pair/trips+. We're also rounding to 6 outs - maybe you have 8 or 9 based on a better draw. Maybe you have 15!

I used FlopZilla (cool tool, BTW, google it) to figure out the percentages of time you graze the flop or better with 67s, which is the hand I had last night. Flopzilla tells me I will hit some part of the flop 29.6% of the time, taking out middle pair or bottom pair (which both give me 5 outs, not quite enough for the shove according to the Ed Miller video). All the other possibilities add up to a rounded 30% of the time.

Using the percentages above, my net result is a loss of $1.50! That also takes into account the pot size at each phase. I assumed a villain's original bet of 3.50. (using .50/$1 blinds), us reraising to $10, and having $60 total stacks.

I tweaked my spreadsheet some and got a positive net winnings by making the preflop fold percentage higher - the breakeven point is the villain folding preflop 63% of the time. This is a more realistic percentage anyway, in my opinion, as long as your own image is tight and you're making this move on the right players, you should expect them to fold to three bets fairly often (until you do it often enough to get them to change their strategy). If your opponent folds preflop 75% of the time, then this play will net you close to 2 BB/hand (1.84). Not a bad profit - better than a blind steal, in fact. (with much more variance, truly).

This calculation was done with 67s. If I improve the hand to 9Ts, I can hit the flop much harder (with top pairs and such) to the point of being able to shove 37% of the time instead of only 30%. This alone improves the winnings to 1.8 BB/hand. If you improve to a "real" hand like KQs, say, then you hit the flop 41% of the time, plus you can no longer assume that you're behind on the flop (your opponent may call you with JJ on a K23 board, for example). So if you bump the winning percentage when called up to something like 45%, no you're earning $2.57/hand. That's a huge amount, if you can take the variance.

If you have any interest in this analysis, let me know via the comments, and I'll share my spreadsheet.