Some Math Stuff For You Guys
While working on some poker scripts for the guys in the #flopturnriver Internet Relay Chat channel, I realized that one particular calculation a lot of new players have trouble with is finding the total value when actions of different values have a different chance of happening. So here I'll give an example that's kind of easy and trivial to show you how to set this type of thing up.
Suppose we're playing heads up no-limit cash and we're on the button. Our opponent has 12bb after posting the blinds and we have him covered. We hold J6s and know our opponent will call a bet with any pocket pair, any ace, any king, any two suited cards, and fold all other hands. Given this information, what would the value of a shove be?
First we open up PokerStove to figure out a few things. What percentage of hands is he calling with? As it turns out, he's calling with 50.2% of hands, so that means he must be folding 49.8% of the time. Now we find out that Q6s has 45.1% equity against that range, meaning that Villain will have 54.9% equity when he does call. Now that we have found all of the needed information, we make a list of all of the possible series of events that could happen after we shove:
A. Villain folds
B. Villain calls, we win
C. Villain calls, we lose
Now for each possible set of events, we figure out the chance of them happening and multiply that times the value we get when that happens. Here I'll break down the three possible scenarios in these terms:
A. Villain folds: We've established that villain folds 49.8% of the time we shove, and we know that will we win 1.5bb (the blinds) when we do, so our value from the times Villain folds is 0.498 times 1.5, or 0.75bb.
B. Villain calls, we win: We already know that villain calls 50.2% of the time and that will we then win 45.1% of the time after that. When we win the showdown, we will pick up 16.5bb (Villain's remaining stack plus the blinds), so our value from these events is 0.502 times 0.451 times 16.5, or 3.74bb.
C. Villain calls, we lose: Similarly to scenario B, we know that villain calls 50.2% of the time, we will lose 54.9% of the time, and when we lose we lose the 15bb we have shoved. Therefore, the value from these events is 0.502 times 0.549 times -15, or 4.13bb.
Now we add up the value that each possible set of events contributes to the overall value of the shove: 0.75 + 3.74 - 4.13 = 0.36bb. So by shoving Q6s against this calling range from Villain with these stacks, we will win on average 0.36 big blinds.
