Originally Posted by BelgoSuisse
When we bet out a hand that has 10% equity on the turn, we win 10% of a $62 pot, i.e. about $6.
When we bet into a hand that is ahead, Tx or 9x, it is typically 95% ahead as we only have 2 outs to a boat, and that means we lose 95% of $28, i.e. $26.
That means we need to fold more than 80% of his range to make betting profitable. Considering this villain calls with any T and any 9 on the flop, i'm not convinced that 80% of his range has neither.
As a sidenote, this is only true if we think it's true that he will never bluff the river after we've checked twice. I've said that he's passive (and I stand by that) but on average our loss will be a little bit bigger because he's going to bluff - or even "value-bet!" - a weaker hand than ours when checked to on the river, even if it's very rare. We run the same risk either way when we check the river, of course, but on average I think his bluffing frequency will be lower then - if nothing else then by virtue of his range being stronger on average, but I think even compensating for that we will still find the number of hands that he'll (rarely) bluff to be smaller.
It's a little more favorable to a bet when you consider he can have draws in his range too, But while decent villains typically have lots of draws in their range as they typically select suited cards and connectors preflop, a 56/7 does not and that unbalances his range towards marginal made hands a lot more than towards draws.
I think you may be either underestimating the number of draws in his range or how much equity they have. My "10%" number from above was not my estimate for what his total equity was for the hands that I don't beat; it was a sidenote on the fact that there IS value in folding out even a hand like A7 (if he should happen to show up with that).
Here's my guess on his range on the turn that I beat:
88-66,44-22,AJs+,Ad8d,Ad7d,Ad6d,Ad5d,Ad4d,Ad3d,Ad2d,KJs+,K8 s-K7s,K5s,QJs,Q8s-Q5s,J8s-J7s,87s,76s,65s,54s,AJo+,A8o,A5o,KJo+,K8o,QJo,Q8o, 87o
It's really hard to guess on what a 56/7 would flat preflop with and then also guess what he folds to a c-bet, but I gave him a 35% range preflop (adjusted with adding some suited connectors that aren't naturally in there and making sure all pocket pairs are in). That range has ~20% equity vs me on the turn and is 250 combos total.
The range that beats me has 95% equity (as you said) and is 128 combos wide (I really wish I had PokerRazor installed on this computer; doing this with Stove took forever). That's including big pocket pairs because I don't know which of those that he 3-bets, but as a compromise I took out AA but left in JJ-KK. So as best as I can see, his range on the turn consists of 1/3rd hands that beat me, and 2/3rds hands that don't. The larger part of his range consists of hands that have on average 20% equity vs me. There are ways for me to make that range smaller (making betting less attractive), but with the unhelpful consequence that its equity goes up (making betting more attractive) so it doesn't seem to matter much.