I have not yet seen the right answer for the first example: AA vs KK. I see a lot of calculations, but they are all wrong.
You have to call an all in on the flop. Now the only thing that matters when talking about EV, is the expected value of the call. The question is correctly asked: "What is the EV of calling knowing we are against specifically KK with our hand, AA?"
And here is where the mistake is made.
Preflop this is the action: - SB posts 0.5 BB - BB posts 1 BB - CO raises 4 BB - SB folds - BB calls 3 BB
So there are 8.5 BB in the pot when the flop comes.
The flop comes 936
And now our opponent will move all in with the kings, and tells us what he have, so we are sure he has KK
We can call or fold. Ofcourse we will call... but what is the EV of this call?
There were 8.5 BB in the pot already, and now the villain put in his remaining 96 BB. This results in a pot of 8.5 BB + 96 BB = 104.5 BB.
Now here is the situation for us: we can call and win, or we can call and lose. Winning will happen 91.6162 % of all times. Losing will occur 8.3838 % of all times.
And now pay attention very closely: When we win, our profit is not 100.5 BB (0.5 BB from the SB plus 100 BB from the villain)! When we win, we will win 104.5 BB. How is that? Well ofcourse we will win the 100 BB from the villain, and the 0.5 BB from the SB... but we will also win the 4 BB that we put in ourselves preflop.
And this is a tricky thing to understand... this 4 BB is ours, so why do we win it? Well, you decided to raise 4 BB, and from that moment on, the 4 BB is not yours anymore, it is in the pot, and you can't take it back. This is a concept that in economics they call "sunk costs" meaning costs that result from decisions from the past, costs that can not be changed anymore. And the 4 BB that you raised are exactly that. They are not yours anymore, so they become a part of your potential profit.
If we lose on the other hand, we will actually be 100 BB down in that hand... but we are talking about the EV of the call here... and the call is 96 BB. So we lose 96 BB on the call, and not 100 BB.
These concepts are VERY important to understand!
So, know that we know that, what is our EV? Well:
EV = (0.916162)(104.5 BB) + (0.083838)(-96 BB) EV = 95.738929 BB - 8.3048448 BB EV = 87.4340842 BB EV = 87.43 BB
And this is the only correct EV from the call. All the other ones you will read in this topic are wrong. On average, you will win 87.43 BB per call you make on the flop in that situation. A very profitable situation as you see, but not as profitable as the author of this topic wrongly suggests: 168 BB.