Calculating folding equity needed in semibluff

[Allinlife]

Pot:36$
Semi-bluff bet: 24$
pot Equity if called 33% with 1 card to come.

I"d appreciate any help I could get /images/graemlins/smile.gif

[thabadguy]

Just by eyeballing it, it doesnt look like he needs to fold for this to be a profitable play, i could be wrong tho.

[olavfo]

To solve this, we have to make some assumptions.

First, let's assume Villain always calls, checks when checked to and folds on the river when Hero makes his hand. The EV equation for betting becomes

EV_bet = 0.33*($36 + $24) + 0.67*(-$24) = $20 - $16 = $4.

On the other hand, if Villain always checks behind and folds on the river when beat, we can make more money by checking, since

EV_check = 0.33*$36 + 0.67*$0 = $12

so checking is a superior line against a Villain that never folds and always checks behind. Even if he sometimes bets when checked to, check/call is more profitable than betting, since we make $12 those times he checks behind, and if he bets, we make $4, just as we would by betting ourselves.

Now assume Villain folds X% of the time when we bet, checks when checked to and folds when beat on the river. How often does he have to fold for betting to be a more profitable line than checking?

The two EV equations now become

EV_bet = X*$36 + (1-X)(0.33*($35 + $24) + 0.67*(-$24)) = X*$32 + $4

EV_check = 0.33*$36 + 0.67*$0 = $12 (like before)

We want to know which value of X makes EV_bet > EV_check, and so we have

EV_bet - EV_check > 0

X*$32 + $4 - $12 > 0

X*$32 - $8 > 0

X > 0.25

So betting is more profitable than checking if Villain folds more than 25% of the time.

To summarize:

If Villain always calls but sometimes checks behind, check/call is better than betting. If Villain bets, it makes no difference, but we make more money those times he checks behind. (This is just another way of saying that you can't bet a drawing hand for value with just one caller when your pot equity is less than 50%.)

Against a Villain that will sometimes fold, and always check behind, betting is more profitable than checking if Villain folds more than 25% of the time.

Note that if Villain don't always check behind when checked to, the equations change, and we get a much more complicated scenario with both the folding frequency and the checking frequency as unknowns. But if we assume "ideal behavior" for Villain, we can solve for the minimum necessary folding frequency and we get 25%.

If Villain sometimes bets and sometimes checks behind, we can of course calculate the minimum folding frequency as a function of the checking frequency, assuming that we know the latter, but I leave this as an exercise for the reader.

olavfo