Ok so last time I posted about an OESFD and how I semi-bluffed pushed the turn with 98 against a LAG. Today, I'd like to take that hand and describe some of the mathematics behind it.

To figure the math, let's give Villain a hand such as AJ, giving him top pair on the flop. I feel it's pretty generous to give him this "big" of a hand considering his 3xBB raising range on the button is enormous. (Recall, he's a 57VPIP/33PFR loose cannon). Also to be exact, our effective stacks were $198.25.

To rehash, I've got 9 outs to the flush and 6 outs to the straight, giving me 15 clean outs and about 56% pot equity. On the flop, I lead for $18 in a $19 pot, and he calls. Now we're on the turn with a board of J T 3 2 and $55 in the pot. I check, he bets $22. So for our purposes now there's $77 in the pot and the action is on me. I push all in for $174.25 (effectively), which is $152.25 more for him to call, which is about 3:2 on his money ($229.25 : $152.25). We know I'm a dog on the turn (35%) so how often must he fold for this play to be neutral EV (break even play)?

Let's figure it out. First let's state in English what our mathematics must prove. There are two results that can happen on the turn, either Villain calls, or Villain folds. When Villain folds, we'll profit $77 (what's currently in the pot) . When he calls, we'll suck out 35% of the time and win the entire pot and 65% of the time we'll lose our investment.

x = % of time Villain folds
(1-x) = % of time Villain calls
currentPot = Amount currently in the pot before we push, or, what we'll win if he folds.
hInvestment = Amount hero risks in order to win currentPot
vInvestment = Amount villain risks in order to win (currentPot + hInvestment)
hEquity = Percentage of the pot hero "owns" over the long haul (our odds of winning).
vEquity = Percentage of the pot villain "owns" over the long haul (his odds of winning).

Ok so we can see our Expected Value will result in:

EV = currentPot * x + (1-x)(hEquity * [currentPot + vInvestment] - [vEquity * hInvestment])

So let's plug in our numbers and figure out how often Villain needs to fold to be EV = 0.

EV = $77x + (1-x)(.35 * [77+152.25] - .65 * $174.25)
0 = 77x + (1-x)(80.24 - 113.26)
0 = 77x + (1-x)(-33.02)
0 = 77x - 33.02 + 33.02x
33.02 = 110.02x
x = 30%

So as we can see, if Villain folds this turn 30% of the time, we break even. Any more than that, and it's money in our pockets. With the given preflop, flop, and turn actions, we've stated that Villain's hand range is extremely wide, and will very rarely have a hand that can stand this much heat.

Let's also take a second to see how some of the different actions could have led to a different outcome of x. Notice if he bets more on the turn, say a pot bet, it will add more dead money to currentPot, and therefore it will cause Villain to have to fold less often for us to profit (23% by my count) than with a weakish bet. However, it is certainly true that his weak bet was what lead me to believe my fold equity (x) was high enough to make this a profitable play.

In the next installment, we'll see what might have happened had we got the money in on the flop. We'll see how our increased equity makes it even better if Villain folds, and how much money we can win by playing our big draws aggressively.

(Note: Special thanks goes out to 2+2's Fimbulwinter (for making the EV calc posts linked below) and also Tom1975 for checking my math).