Suppose you have 6d7d on the button, and the flop is 7s5h4d, and the pot is 48.5 bb. Clearly, we have top pair, and the outs we have to improve our hand are 2 7's (for a three or four of a kind), 4 3's, 4 8's (to obtain a straight), and 3 6's (for two pairs), and one out for a flush draw. Using approximate probability calculations you have around a 56% chance of winning the hand. Now your opponent (playing at CO) bets 6.5bb, which gives you an expected value of (55)(.56)-(6.5)(.44) = + 2.84 so it makes sense to call.

Now the turn is a Qs, and your opponent bets 33.5 bb. We still have the same outs with the exception of the flush, and multiplying these outs by two we obtain 26% chance of winning. Since the pot is now 95 and the bet is 33.5, our expected value is (.26)95-(33.5)(.74) = -.09. Now this leads me to believe that the correct move would be to fold especially since your opponent could have a queen, and you are putting in quite a large bet to continue. According to the GTO program my EV was positive to call so I'm a little confused as to how they arrived at that value.

Clearly there are more things going on than just the basic probabilistic definition of expected value in these calculations. Without understanding how the program is computing these expected values I'm unsure as to how I'm supposed to learn from the mistakes I make.

So my questions are:

(1) Is there a source which explains how these programs compute expected values?

(2) If not, how is one supposed to learn from working with these GTO programs?


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