Quote Originally Posted by MadMojoMonkey View Post
The equation may make more sense if presented like this:

EV = (fold%)*(18.5) + (1-fold%)*(something)

something = (equity%)*(105.5 - rake) - (1-equity%)*(96)

EV = (fold%)*(18.5) + (1-fold%)*( (equity%)*(105.5 - rake) - (1-equity%)*(96) )

EV = (fold%)*(18.5) + (1-fold%)*(equity%)*(105.5 - rake) - (1-fold%)*(1-equity%)*(96)


All I did was show why in the hell the (1 - fold%) term shows up twice.
Quote Originally Posted by acg123
I see it showed up twice, and this might be a dumb question but why does it show up twice?
Start here:
EV = (fold%)*(18.5) + (1-fold%)*(something)

Either he folds, or he "not-folds." In this case, since he's responding to a shove, there is only 1 "not-fold" option, so we don't need to separate call and raise.

It would normally start like this
EV = (fold%)(value_fold) + (call%)(value_call) + (raise%)(value_raise)

Any of those variables in parentheses can be 0, eliminating that term from relevance.
In this case, Her has gone all-in, so Villain can not raise. This makes our equation look like this:
EV = (fold%)(value_fold) + (call%)(value_call) + (0)(value_raise)
EV = (fold%)(value_fold) + (call%)(value_call)

We know that our equation is incomplete if the %-ages don't add up to 100%, or 1.
So we know that
fold% = (1 - call%)
and also that
call% = (1 - fold%)
Renton picked fold% to work with and eliminated one variable for his choice.

So now the EV euqation (when Villain is responding to a shove) looks like this:
EV = (fold%)(value_fold) + (1 - fold%)(value_call)

So now we know the value of a fold (the dead money in the pot before Hero shoved) and we know (or hypothesize) Villain's fold%.
The thing we need to solve for now is
value_call = something

Well... We can work out the something... since there are no more bets allowed at after this action, that makes it purely an equity analysis.
something = EV when called = (equity%)*(value_win) - (1-equity%)*(value_lose)

We use the same principle above and assume there will be no ties (which gives the mathematically same answer as if there ARE ties, so we're not even approximating, here).
That allows us to shorthand equity%_Hero in to just equity%, since we know that either Hero wins or Hero loses, so only Hero's equity is important. We could get the same results by using Villain's equity, but we're looking at it from Hero's perspective, so we'll choose Hero's equity.

OK, taking it back. Now we have this:
EV = (fold%)(value_fold) + (1 - fold%)(value_call)
and this:
value_call = (equity%)*(value_win) - (1-equity%)*(value_lose)

So we substitute the 2nd equation into the 1st.
EV = (fold%)(value_fold) + (1 - fold%)( (equity%)*(value_win) - (1-equity%)*(value_lose) )

The (1-fold%) term distributes through the 2 terms in the (value_call) substitution.