EV calcs give me a headache usually so bare with me. Will also ask someone to have a second look.

The approximate range I think he gets to the river with (intenationally made it 100 combos for simplicity but we can add value/bluffs it would get to the same ratio imo and from initial post I think you agree with the range)

**Board: **T
T
Q
2
2

** Equity Win Tie**

**MP2 ** 55.95% 52.95% 3.00% { KcQd }

**MP3 ** 44.05% 41.06% 3.00% { JJ-77, AQs, ATs, KTs+, Q8s+, J9s+, T7s+, AQo, ATo, KTo+, QTo+, JTo, Jh9d }

I assume he folds 41 combos of air and midpairs so folds 41% of the time when we bet.

**Board: **T
T
Q
2
2

** Equity Win Tie**

**MP2 ** 100.00% 100.00% 0.00% { KcQd }

**MP3 ** 0.00% 0.00% 0.00% { JJ, 99-77, KJs, QTs, J9s, KJo, Jh9d }

EV(fold) = 22.20 => 0.41*22.20 = 9.1

Calling range when you bet:

**Board: **T
T
Q
2
2

** Equity Win Tie**

**MP2 ** 57.66% 46.11% 11.55% { KcQd }

**MP3 ** 42.34% 30.80% 11.55% { AQs, KQs, Q8s+, AQo, KQo, QJo }

26 combos of that.

EV(call) = 36.2*0.576 = 20.85

26 combos = > 5.42

Raising:

**Board: **T
T
Q
2
2

** Equity Win Tie**

**MP2 ** 0.00% 0.00% 0.00% { KcQd }

**MP3 ** 100.00% 100.00% 0.00% { TT, ATs, KTs, QTs, JTs, T7s+, ATo, KTo, QTo, JTo }

EV(raise) = -7 => 0.33 * (-7) = -2.31

EV = 9.1 + 5.42 - 2.31 = 12.21

If sizing doesn't change his calling and raising range, your EV will remain the same.

If you check, you check/call against his polarized range. First of all, you win when he checks a certain amount of the time:

**Board: **T
T
Q
2
2

** Equity Win Tie**

**MP2 ** 77.97% 71.96% 6.01% { KcQd }

**MP3 ** 22.03% 16.02% 6.01% { JJ, 99-77, AQs, KQs, QJs, Q9s-Q8s, AQo, KQo, QJo }

He checks 50% of the time.

EV(check) = 22.20 * 0.78 = 17.31 => 0.5*17.31 = 8.65

There are 50 combinations left in his range, some nuts and some air. He has 33 nut combos which are 66%.

Nuts:

**Board: **T
T
Q
2
2

** Equity Win Tie**

**MP2 ** 0.00% 0.00% 0.00% { KcQd }

**MP3 ** 100.00% 100.00% 0.00% { TT, ATs, KTs, JTs, T7s+, ATo, KTo, QTo, JTo }

EV(betnuts) = -15.20 * 0.66 = - 10.03

This means you need to make up for 12.21 < 8.65 - 10.3 + EV(betair).

12.21 < -1.65 + EV(betair)

EV(betair) > 13.65 so that check/calling is better than bet/folding.

Let's see what EV(betair) looks like.

EV(betair) = x (unknown frequency) * 52.6.

13.65 = 52.6x => x = 31.6% so we need him to bluff almost ALL his busted draws since there are 33% left in his bluff combos. That's just to be BE. If he bets larger we get worse odds so we need more bluffs if he bets smaller we need fewer bluffs.

Hope I didn't screw up (too much, there are bound to be some small mistakes here since I'm actually coming down with a headache

). I'm not much of a maths guy in such very very close spots because what maths will tell you is that both action are very very close in EV but it mostly depends on your opponent.

What I recommend is that whenever checking and betting is close in EV, just bet, get aggressive, you get a better image and they might actually call worse than you expect when you vbet so thin, because they look at frequency and forget you can be polarized or depolarized with your frequency.

I'm off to bed now! GOOD NIGHT!