Hoping for some help. What is the mathematical formula to find how often villain must fold to our bet according to bet ratio to pot.

It doesn't have a great deal of practical use, but basically you need to work out the amount we lose when called, which is:

size of bet * percentage of times villain calls * percentage of times villain wins when calls

and then work out the bolded so that the above is equal to:

(size of pot * percent of times villain folds) + (size of pot + size of bet)*(percentage of times villain calls)*(percentage of times villain loses when calls)

Or something like that. That's assuming it's an all in bet, if there's money to be bet on future streets it becomes even more complicated

Yes, it's a moving target ... Your factors are % of calls and V's calling range with both of these being pretty difficult to pin down. And those can change depending on which street of betting you are on as well.

But in the simplest terms ... Pot = 100, You bet 50 ... 150/50 is 3. So V must fold 1 out of 3 tries to break even. (This assumes you are behind 100% of the time when called)

To move this in your favor, assuming V still calls 67%, just change the bet a little. Pot = 100, You bet 40 ... 140/40 is 3.5. Now you just swung it in your favor by 20 chips net.

V calls, you lose 40
V calls, you lose 40
V folds, you win 100 ... net gain of 20 chips over folding/showdown loss


Let's change the scenario a little to see what happens ...

Checks to Showdown .. Hero wins 25% ... win 25 chips (100 pot x 25%) at zero additional risk


Pot = 100, Bet = 40, V = 100% call, Hero win = 25% ... win 45 chips (180*.25), but you had to risk 40 'more' to do so .. thus ...

Lose 40 3 times + win net 140 once = win 'only' 20 more net by risking 40. Not a good long term plan. GL

The basic fold equity calculation is quite simple.

Fold Equity required (when bluffing with air) = size of bet / (pot size before the bet + size of bet)

In simple notation, FE = b/(p+b)

E.g. If the pot is 100 and you bet 50 as a bluff, you need villain to fold 50/(100+50) = 1/3 of the time.

If maths is hard for you, you can just memorize some numbers.

Pot size bluff => Villain must fold 50% of the time.
3/4 pot => 43% fold equity required.
60% of pot => 37.5% fold equity.
1/2 pot => 33.33% FE.
1/3 pot => 25% FE.

Extrapolating from that to use ratios, you should be able to see that if your bet = 1* pot, such that the ratio of bet to pot is 1:1, then villain needs to fold 1 out of 2 times.
1:1 bet => fold 1/2
1:2 bet (half pot) => fold 1/3
1:3 bet (third of pot) => fold 1/4

See the pattern?

It’s not quite as simple as Arty showed if your bet has some equity, which is usually the case. The equation is:

FE= (Bet -X)/(Bet + Pot) - X),

where X= eq*(Pot + 2*Bet)

Example: You have a turn flush draw. Assuming you win with a river hit, your equity is 0.20. The pot is 3 and you go all-in with a bet of 4. The fold equity you need is gotten as follows:

X = 0.2*(3 + 2 * 4) = 2.2

FE = (4-2.2)/(4+3-2.2) = 1.8/4.8 = 37.5%