Required overlay for flush draw on paired board

[also]

I was an observer of the following hand, and afterwards I criticized the hero's play. Well, he was right, I was wrong.

Fairly loose, passive home game. 5 players limp, so 7 see a flop of T62, two hearts. The flop is checked around, and the turn is (T62)T, two hearts and two spades. The big blind bets, a loose player calls, and our hero calls with a nut flush draw (he didn't say, but I'm assuming spades), getting 5.5:1 on his money.

Our hero's thinking is that the checked flop made a ten unlikely, so a good part of the time the board pair is irrelevant to his drawing chances. I pointed out that the big blind is more likely than any other player to hold a weak ten and check it on the flop (he is also more likely than the others to flop a trashy two pair with that board and try to check-raise), so 5.5:1 on hero's probable 7-out shot when he might be drawing dead is insufficient. Let's have a looksie.


Hero is getting 5.5:1 on the turn to call. Let's assume the bettor always bets the river, whether he is bluffing or not, and that our hero calls on the river if he hits his flush and folds if he misses.


If the bettor is already full, then hero's call EV is:

9/45 * -2 + 35/44 * -1 = -1.20

If the bettor has a ten, then hero's call EV is:

7/45 * 6.5 + 36/45 * -1 + 2/45 * -2 = .122

If the bettor is bluffing, then hero's call EV is:

9/46 * 6.5 + 35/46 * -1 = .511


Now, let's try to figure out the chance that our villain has trips vs. a boat/quads on the turn, given that he is not bluffing. He was in the blind, so let's assume all hands are equally likely. On a board of TxyT, he can hold:

xx: 3 combos
yy: 3 combos
TT: 1 combos
Tx: 6 combos
Ty: 6 combos
----------------------------
total boats/quads: 19 combos

If he has just trips (Tz), then there are 38 possible kickers (everything except the 7 cards we've seen and the 3 y's, 3 x's, and 1 ten). There are 2 * 38 = 76 trips combos he could hold.

Given that he has at least trips, then, our villain holds a boat there 19/(19+76) = .2 of the time.


In the cases where our villain is *not* bluffing, hero shows a profit of:

(.2 * -1.20) + ((1 - .2) * 1.22) = 0.736

If we mix in the bluffing case, the expectation can only improve. So, in conclusion, it's a super solid call. A big part of the reason I thought it was a bad call was that during the hand, the 6 outs odds (~6.5:1) popped into my head rather than the 7 outs odds. (Yeah, I'm mister super pro. I'm mister pot odds expert. Hahaha.) (A 6.5:1 draw is definitely *not* a call in this spot, BTW.)


This analysis ignores a few details. For instance, I assume no one behind our hero *ever* raises. This is somewhat reasonable given the fact that the players in position didn't bet the flop and that the pot is now protected, but "never" is a big word. At least occasionally someone in position will have flopped a set or ridiculous two pair and decided to slowplay. I also assume that, in the "bluffing" case, neither of our opponents holds a random pocket pair, which would reduce hero's outs from 9 to 8.