Originally Posted by CallMeVernon
Value and Equity
Let’s go back to an example from before (slightly tweaked for simplicity): we have KK, top set, the board is K84rb, and the pot is $100 with $475 left to bet. Let’s say that if we bet $25, our opponent will call with a range of all no-gap connectors that have one pair or a gutshot, meaning 98/87/76/65/54. (I’m narrowing it down to this because we’re going to assume this is the part of his overall range that wouldn’t also call a bigger bet.) We are making this bet intending to string our opponent along so that he can “get there” on the turn, when we will make a bigger bet that will price out all hands that have not improved. (Ignore things like improving to a pair plus gutshot—pretend we’re going to price that out too.)
First, let’s count combos. There are 12 combos each of 98, 87, and 54, for 36 combos of 1 pair hands that have 5 outs to improve (and for simplicity we’ll assume they’re actually drawing dead and ignore runner-runner quads). There are 16 combos each of 76 and 65, for a total of 32 combos of hands that have live equity against us. So 32/68 of the time, our opponent is drawing profitably against us, and 36/68 of the time, our opponent is drawing dead against us.
Next let’s figure out, using our opponent’s perspective, what his EV is by calling with the part of his range he is correct to call with—the gutshots. A gutshot will have 4 outs to beat us, with 45 unseen cards. So 41/45 of the time, he loses 25. (4/45)(34/44) of the time, he hits on the turn and stacks us (and gets the $100 pot). (4/45)(10/44) of the time, he hits on the turn and gets stacked when we hit our boat redraw. Villain’s EV is
(451/495)(-25) + (10/495)(-475) + (34/495)(575) ~ 7.12
Now let’s figure out Villain’s EV from drawing dead against us. He has 5 outs to get stacked, and if he misses, he gets away only losing $25. So his EV is
(1/9)(-475) + (8/9)(-25) = -675/9 = -75
Finally, Villain will have the gutshot only 32/68 of the time, and he is drawing dead 36/68 of the time. So Villain’s overall EV is
(8/17)(7.12) + (9/17)(-75) ~ -36.35]