Small PP vs 3bet ~75bbs deep

http://forumserver.twoplustwo.com/185/heads-up-sng/small-pp-vs-3bet-75bb-deep-1048862/

Poker Stars, $xx Buy-in (10/20 blinds) No Limit Hold'em Tournament, 2 Players
Poker Tools Powered By Holdem Manager - The Ultimate Poker Software Suite. View Hand #9651022

BB: 1,440 (72 bb)
Hero (SB): 1,560 (78 bb)

Preflop: Hero is SB with 5 5
Hero raises to 50, BB raises to 160, Hero ???




Air-Bear:
The bigger villain's 3bet-fold range is, the better shoving becomes (this is obvious). The variables we should consider are villain's 3b-fold percentage and our equity vs villain's range.

The general formula to calculate your EV in this spot is:
EV = P(villain calls shove)*(equity* [chips we can win] - villain's equity* [chips we can lose]) + P(villain folds to shove)*(dead money) 
note: P(...) stands for "the probability of (...)


Now let's unleash the maths:
Imagine that you know your opponent will stack a range of 9% (A9s+, ATo+, 77+). We can calculate how much the villain should be 3betting (eg how big his 3bet-fold range is) to make a shove +EV.


Villlain's 3bet range is our variable, let it be X
P(villain calls shove) is 0.09/X
Our equity is about 39% when called
Villain's equity is 61%
We can win 74.5 bb from villain (effective stacksizes+our dead money)
We can lose 69.5 bb (72-2.5)
P(villain folds to 3B) is 1-(0.09/X)
dead money = 10.5 bb (2.5+8)

Now just fill in the info and solve the equasion for X:
EV = (0.09/X)*(0.39*74.5-0.61*69.5) + (1-(0.09/X))*10.5)
EV = (0.09/X)*(-13.34) + 1*10.5 - (0.09/X)*10.5
...
EV=0 for X~= 20.5%. So if villain's 3betting more than 20.5% (and calls the range I gave) it's a +EV play.

I hope my math is right, would be cool if someone could check it quickly

Just play around with the variables if you want to know your EV for different situations.






I understand the concept above intuitively but knowing how to do the math and playing around with the numbers is something I don't do enough of. I need to spend half a day or so with excel playing around with equities and then make a post to solidify what I learn.