Quote Originally Posted by elipsesjeff
This is probably something you've done before so it shouldn't be that bad .

Calculate the total EV of calling a preflop re-raise with 55. We raise UTG 6 handed at 25 NL to $.75 and get re-raised to $2.50 by the button. Lets say he has QQ-AA and AK in his reraise range in this spot. Effective stacks were $25 to start the hand.

In a re-raised pot we will get AA to stack off 100% of the time on any board, KK 100% on any undercard boards but only 1 3/4 size pot bet on an Ace, AK 100% of the time when he flops TPTK or better and we flop our set, and QQ 50% when he flops an overpair and again only 1 3/4 size pot bet on an Ace or King. What's the total EV of calling?

Follow up question: If we include JJ/AQ in the above reraising range, and add in JJ stacking off 33% on an undercard board and only 1 1/2 pot bet on an A/K/Q. AQ you get 2 1/2 pot bets, but not his stack when he flops TPTK or better when we flop a set. Does this change the EV enough to swing the decision one way or another?

3rd Question: If villain's only reraising with AA/KK here can we call pre-flop, given that he stacks off with AA preflop 100% and KK as an overpair 100% and 1 3/4 psb when an Ace comes?
These are all pretty straight-forward EV equations. Instead of answering them individually (which won't be of that much use to most people who read this thread since they'll see a wall of text with some math in it, shit themselves, and move along) I'll give some tips on making these long EV equations as painless and mistake-less as possible.

The first thing you should do is list out all of the possible outcomes. This can be done with the assistance of a diagram or whatever you need for it. The third question has the fewest outcomes so I'll do it as an example:

1) We miss the flop
2) We hit the flop, Villain has AA, Villain does not outflop us, we win showdown
3) We hit the flop, Villain has AA, Villain does not outflop us, we lose showdown
4) We hit the flop, Villain has AA, Villain does outflop us, we win showdown
5) We hit the flop, Villain has AA, Villain does outflop us, we lose showdown
6) We hit the flop, Villain has KK, Flop comes A-high, Villain does not outflop us *
7) We hit the flop, Villain has KK, Flop comes A-high, Villain does outflop us, we lose showdown
8) We hit the flop, Villain has KK, Flop comes A-high, Villain does outflop us, we win showdown
9) We hit the flop, Villain has KK, Flop does not come A-high, Villain does not outflop us, we win showdown
10) We hit the flop, Villain has KK, Flop does not come A-high, Villain does not outflop us, we lose showdown
11) We hit the flop, Villain has KK, Flop does not come A-high, Villain does outflop us, we win showdown
12) We hit the flop, Villain has KK, Flop does not come A-high, Villain does outflop us, we lose showdown

* We'll assume that when he has KK and the flop comes A-high if he misses a set that he bet/folds the flop so that we don't have to deal with the times we see a turn and he hits his set and puts more money in, etc.

The total EV of the situation will be the sum of the individual EVs of the outcomes. The EV of each individual outcome is the chance it happens times our profit for that outcome (including our call preflop). I'll work out the individual EVs of two of these outcomes, an easy one to understand the idea and a hard one to understand the detailed process.

1) We miss the flop

There are 48 cards left in the deck and two of them make our hand. The chance of none of our cards coming on the flop is (46/48)(45/47)(44/46) = 0.878 or 87.8%. That is the chance of this outcome happening (note that this also means the chance of at least one 5 coming is 1 - 0.878 = 0.122 or 12.2%, which we'll use later). Our profit for this scenario is -$1.75 since that is the amount for our call preflop. Since the EV of each individual outcome is the chance it happens times our profit for that outcome, the EV of this particular outcome is 0.878 * -1.75 = -$1.537.

9) We hit the flop, Villain has KK, Flop does not come A-high, Villain does not outflop us, we win showdown

Note: I'm going to bold a few numbers as we go down because in our final calculation for this outcome I don't want people to be thinking "Holy shit where did this number come from?"

The chance Villain holds KK is 50% since his range is {KK+} evenly distributed. Let's find the chance that at least one 5 comes on the flop, the flop does not contain an Ace, and the flop is not K5x (where x is not a 5). This is the part where peoples asses usually start churning butter milk, but it's not that bad really.

First, the chance the flop comes K5x where x is not a 5 is (2/48)(2/47)(45/46) * 6 = 0.0104 or about 1.04%.

Now what's the chance total that the flop comes with at least one 5 but without an Ace? The chance of a 5 coming on the first card is 2/48. The chance of an Ace NOT coming on the second card is 43/47. The chance of an Ace NOT coming on the third card is 42/46. Then the total chance of this type of flop is (2/48)(42/46)(43/47) * 6 = 0.2088 or 20.88%.

Then the chance of a flop coming with at least one 5, no Ace, and not K5x (where x is not a 5) is 20.88 - 1.04 = 19.84% or 0.1984.

Once we see this flop, what is the chance we win a showdown? It's going to be roughly our equity on the flop, so we'll use that, which is about 91% give or take some runner draws for KK.

Our profit for this outcome is that we win what was in the pot before we called plus the rest of our opponent's stack. The pot when it is our turn to act is $3.60 and the remainder of our opponent's stack is $22.50. This assumes 0.10/0.25 blinds (some sites use 0.15/0.25 instead) and no rake. Accounting for the rake in these types of calculations is pretty easy, you just subtract it from the profit in the outcomes where you win the hand. So our total profit for this outcome is 3.60 + 22.50 = $26.10.

Then the final calculation is easy. Recall our outcome: We hit the flop, Villain has KK, Flop does not come A-high, Villain does not outflop us, we win showdown.

- We hit the flop and it's not A-high and Villain doesn't outflop us: 19.84%
- Villain holds KK: 50%
- We win showdown: 91%
- Our profit: $26.10

So the total EV of this outcome is 0.1984 * 0.5 * 0.91 * 26.10 = $2.356.

This goes a lot faster when you use a spreadsheet and some sort of chart or program to give you the flop/turn/river and equity calculations.