TPMK type hands HU

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From PokerStove:

Board: Ac 9d 6h
equity(%)win(%) tie(%)
Hand 1: 54.5% 48.32% 06.15% { ATo }
Hand 2: 45.5% 39.39% 06.15% { AA, A2s+, A2o+ }

So if your only assumption is that opponent has an A and ignore all information from opponent's betting actions, then you are a solid favorite to be ahead at the river.

But if the turn is a small card, you are no longer a favorite:
Board: Ac 9d 6h 3s
equity (%) win (%) tie (%)
Hand 1:49.9% 46.52% 03.38% {ATo}
Hand 2:50.1% 46.72% 03.38% {AA, A2s+, A2o+ }

And if the river is another small card that misses you, you are a big underdog:

Board: Ac 9d 6h 3s 8s
equity (%) win (%)tie (%)
Hand 1:40.3% 36.80% 03.45% { ATo }
Hand 2:59.7% 56.29% 03.45% { AA, A2s+, A2o+ }

I tried several different small cards for the turn and river and the results were almost identical. So I guess- No you should not bet the river if you know your opponent started with Ax and nothing in his betting suggests that he missed.

Interesting.

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Wow, thanks for all the leg-work you did in helping solidify my observation. Good job!