I'm not good at math and I'd like some help on finding a formula to use to solve this problem.

Let's say I want to know what % of the pot someone needs to bet in order to give me proper odds to call if I have x% of equity in the hand, then what would be the formula?

Thanks

First, this type of thinking only works on the river, because future streets can impact your true equity. This is poker and not math related, but for example say an opponent either has the nuts 50% or a bluff 50%. If he bets $0.50 into the $1 pot with his entire range, and there is only one street of betting, you can easily call because he is bluffing half the time and you only have to be right 25% to call. But if there are two streets of betting, you probably have to fold, since on the next street he can bet much bigger.

Now to the formula. If they bet p of the pot, then you have a price of (p/(1+2p) ). So you should call if x >= p/(1+2p).

For example, if they bet half the pot, p = 0.5, and you should call if x >= 1/4.

Lets move this to poker theory or something. And maybe even change the title maybe to "How much of pot to call as function of your Equity" if OP agrees for better search results.

Clearly if you have equity 50% or more you can call whatever they bet without any problem, the more the better of course.
If your equity is less than 50% then you need to call an amount x when the pot was y so that the resulting pot (x+x+y) times your equity is larger than what requested to pay x; or

E*(2*x+y)>x

or

x<E*y/(1-2*E)

So as fraction of the pot you can call up to E/(1-2*E). with E your equity.

Of course that assumes there is no further betting say river Equity estimation. If there is further betting eg that was a turn (or worse even flop bet) and at river you hit you can probably get a bit more or even more rare lose a lot if they have upgraded too to a higher hand (implied and reverse implied odds).

So in practice if there is possibility of future betting and you are drawing to near nuts you can call even more than the above as long as its not a lot more.

So use the above expression for no more betting cases and adjusted higher for implied odds of more bets that can be possibly called when you make your big hand (the higher they call at river the higher your tolerance for turn calling becomes obviously)


Example with a flush draw and mid pair and maybe an overcard as possible outs -although not exactly clean- against someone that has shown strength and is not often at all bluffing at this spot due to preflop position and action and overall image say, you seem to have equity about 28.9% eg say some As7s in a 2s Ts 7h 3c board so it feels that opponent is either on overpair or maybe AT or set or a rare bluff or a rare other draw (say not often due to position he had preflop) so ignoring future bets (as said not a good idea in general but enough of a restriction to see how bad a call is with plenty of room for error) the above formula gives;

0.289/(1-2*0.289)= 68.5%

so you can call up to 68.5% of the pot. Basically with implied pot odds ie future bets its probably ok to go for even 70-75% of pot and with some opponents that are not very tight and careful even pot bet but no more unelss you have strong confidence they will give you some action at river even if little.

If you had on the same board only an 8c9c say for a straight draw typically only (meaning opponent can now have also top AsXs flush draws since you no longer have the ace) your equity would be only 17.9% and some of it possibly not even realized at river because if you connected an 8 or a 9 vs him having air with some say AK busted draw its not sure you are going to win the pot anyway as you probably cannot catch a bluff easily. That smaller equity now forces you to;

0.179/(1-2*0.179)=28%

So you can tolerate only some 28% or if future bets involved probably around 40-50% of pot bet no more.


To better see what i am talking about regarding say river bets if the discussion was about a turn call imagine that if you connect they almost always will at least call or even better bet themselves some pot/3 at river or better (lets say this balances the rare cases they dont bet or call anything or they bet a bit more than pot/3 and we raise and they fold or call and they lose or they raise you even or bet a lot more than pot/3 and you win or they check-raise or bet-reraise you and you call and lose because they have say fh or some better hand in general now etc) then above thinking becomes;

If you call x at turn on a pot y you lose that x the (1-E) fraction of the time you do not upgrade to your draw to the better hand you wanted and win x+y+ say 1/3*(2*x+y) additional river bet (the pot/3 avg we set as reasonable avg target) the E fraction of time that you win the hand. This of course assumes you cannot win with a bluff when you fail to connect (or with a check/check even) which if you are a good player vs a weak or careful opponent and you bluff makes sense it might still be a small avg source of profit.

In any case the above assumptions lead to; (in order to show a profit by calling that turn x bet)

E*(x+y+1/2*(2x+y))-(1-E)*x>0

or

x<4/3*E*y/(1-8/3*E))

So feel free to call up to say 4/3*E/(1-8/3*E) at turn.

That means if you had the above straight draw for example E=17.9% giving
45.6% pot bet as threshold.


Look back at the flush plus pair draw of the 28.9% equity example and we obtain even a massive 1.68 times pot bet as threshold.

Basically the stronger draws when you have at least 22-29% equity imply that with a little bit of river action even if not a lot strong action when you connect you can tolerate up to pot bets at turn and survive just fine. Of course the smaller the bets you call at turn and the higher the bets you extract when you connect at river the better it will be so we must not lose touch with the requirement to get as cheap as possible to where we have less than say 30% equity (with future bets coming) or 50% if no more bets coming.

Thank you very much for taking the time to answer in such great details. I really appreciate it.

And of course if you want to move it to Poker Theory and change the title it is perfectly fine, your post could help more people this way.

I think a mod can do this? If it's my responsability just tell me how to do it and I'll be happy to change it.