tough spot. i suck at this kinda stuff but am trying to get better at it. let me know if you see any mistakes.

what range did you put him on after he called the flop?

i'd probably say: Kx, Jx, QT, pocket pairs like 66-TT, possibly a slowplay with KJ, JJ, or 55, or some weird gutshot like AT, Q9, AQ, etc...

the only hands out of that range that would check the turn through are jacks, gutshots that didn't fill in, and small pocket pairs.

the river really fucks you pretty hard here and i can't see much you're ahead of.

say 44 - TT = (excluding 55 which would have bet the turn)

6 combos 44
6 combos 66
6 combos 77
6 combos 88
6 combos 99
6 combos TT

reasonable jacks

J9 - 8 combos
JT - 8 combos
QJ - 4 combos
KJ - would have bet the turn
AJ - would have bet the turn

42 you beat
20 you lose

considering how ugly that board is for 44-TT against a preflop 3bettor, i think most of these hands are looking to get to SD here cheaply. how often they call a bet here is what's really important.

i think if you check/call here, i'd estimate you're beat like 85 - 90% of the time.

42 you beat, 20 you lose.

if you played out this river 62 times.

20 times you'd bet/fold to his raise for - 20bb

the other 42 times, he'd have to call a bet more than 20 times in the 42 shots to show a profit. 20/42 = 47%, which i think is doable.

i think this is a bet.

let me know if i made some mistakes? thanks.