semi-bluff maths help pretty please

ok so i'm reviewing some hands and one came up where i thought this piece of knowledge, and specifically how to acquire it, would be very helpful for me.

basically i want to know how to determine the required amount of pot equity for all in semi-bluffs to be neutral EV based on certain amounts of fold equity. for the sake of discussion, here's the hand and risk/reward amounts simplified.

PokerStars No-Limit Hold'em, $0.10 BB (5 handed) - PokerStars Converter Tool from FlopTurnRiver.com

UTG ($20.38)
MP ($10.10)
Button ($10)
Hero (SB) ($14.52)
BB ($9.25)

Preflop: Hero is SB with two cards.
2 folds, Button bets $0.30, Hero raises to $1.05, 1 fold, Button calls $0.75

Flop: ($2.20) 8:club:, 5:club:, 4:spade: (2 players)
Hero bets $1.20, Button raises to $3.45, Hero raises to $13.47 (All-In), Button calls $5.50 (All-In)

pot at point of hero's shove: $2.2 + $1.2 + $3.45 = $6.85
hero's risk is $7.75 (how much villain has left in stack + how much he raised me by)

is there some equation we can set up to determine what the minimum amount of equity i need for a shove to be neutral EV if villain folds x% of the time? for posterity, let's say i estimate my opponent will fold 10% of the time. what is the easiest way to determine the exact amount of pot equity i need for a shove to be 0EV (and hence, the bottom of my shoving range)

thankyou for any responses.

Ev[shove] = (%call)(Ev[call]) + (%fold)(Ev[fold])
Ev = (0.90)([pot equity](total pot) - (amount shoved)) + (0.10)(6.85)
0 = (0.90)([X](20.10) - 7.75) + 0.685
0 = (0.90)(20.10X - 7.75) + 0.685
0 = 18.09X - 6.975 + 0.685
0 = 18.09X - 6.29
6.29 = 18.09X
X = 0.3477

So 34.77% equity needed with 10% fold equity. Also you only need around 38.6% equity to have a breakeven shove here if you know villain always calls.

If we assume villain has a range of something like (44,55,88+,76s, JcTc, QcJc, KcJc, KcQc, AcJc, AcTc, AcQc, etc) we would have the necessary 38.6% equity to shove with a range of (that would could reasonably 3bet with preflop):

QQ+, 8d8h, 8d8s, 8h8s, 5d5h, 5d5s, 5h5s, 4d4h, 4d4c, 4h4c, 76s, AcKc, AcQc, KcQc, AcJc, KcJc, QcJc, AcTc, KcTc, Ac9c, Kc9c, Ac7c, Kc7c, Qc7c, Jc7c, Tc7c, 9c7c, Ac6c, Kc6c, Qc6c, Jc6c, Tc6c, 9c6c, 8d5d, 8h5h, 8s5s, Ac4c, Kc4c, Qc4c, Jc4c, Tc4c, 9c4c, 8d4d, 8h4h, 7c4c, 6c4c, 5d4d, 5h4h, Ac3c, Kc3c, 7c3c, 6c3c, 4c3c, Ac2c, Kc2c, 7c2c, 6c2c, 4c2c, 3c2c

that post has hit the nail on the head. thanks alot, stacks

Thanks a lot for this Stacks (and rpm for original question) - this formula and playing around with the variables has really got me thinking a lot.

This a great question and even better answer (stacks), will definitely use. :goldstar: