Home Game - Friendly

$.50/$1 - 4 Players

CO: $150
BTN: $150
SB: $150
BB (Hero): $150

Pre-flop:
CO opens to $2
BTN folds
SB calls $1.50
Hero raises to $6 with AQ
CO calls $4
SB folds

($14) Flop: AJ10
Hero bets $7
CO calls $7

($28) Turn: K
Hero checks
CO bets $14
Hero raises to $30
CO raises to $137 and is ALL IN

The thing is, when CO shoved he turned over his hand and showed KQ, obv doesnt want to chop.

What course of action should I take on the turn?
My pot odds do outweigh his outs on the river so this is a call, I think.
Check/call turn? Then check river?

Kinda need reads on the players to decide whether of not to 3bet or flat pre.

If you decide to 3bet raise bigger and bet more OTF.

Definitely bet the turn. You don't want to give free cards to two pairs, flush draws, and although unlikely, sets.

As played, you have a math problem to solve.

Bet the flop bigger.
We have a pretty good hand here, there are lots of worse hands that we can get value from including worse Aces, flush draws, worse 1p hands, pair + flush draw, pair + straight draws and lots of other stuff.

On the turn, just keep betting. When you check/raise you are pretty much screaming, I have the nuts, and you are also opening you self up to the chance that he will check behind and be able to see two cards for the $7 price that you set for him on the flop.

As for what you should do now, do some math.
80% of the time you chop, and 20% of the time you lose.
Then you will have you answer.

Bet turn, don't fold now.

So you're calling $107 to win... ~$45ish?

So you're giving around 1-2.5 to make the call, but he's only gonna make his flush around 20% of the time.

It's a call.

If you call,

80% of the time you win 50% of the final pot (you lose 100% of your final bet 20% of the time).

Pot will be $302, and it's $107 for you to call.

Chop EV: 80% * ($302 * 50% - $107) = $35.2.
Lose EV: 20% * -$107 = -$21.4.

Looks like total EV of a call is positive $13.8 = $35.2 - $21.4.

Is my math right? Is there a more intuitive way to look at this?

OK I'm actually gonna work on the math here to figure the break-even point.

Some algebra.

X = pot.
Y = bet.

Chop: 80% * (X * 50% - Y) = EV of call and chop
Lose: (20% * -Y) = EV of call and lose

Now I want to find the break even point whereby

EV of call and chop = EV of call and lose (we eliminate negative sign here, yeah?)

Setting that up:

80% * (X * 50% - Y) = (20% * Y)

0.8 * (0.5X - Y) = (0.2Y)
0.4X - 0.8Y = 0.2Y
0.4X = 1.0Y

So that's saying the pot has to be 2.5x the bet to be profitable.

So here, the pot is 302 / 107 = 2.8x times the bet. Should be profitable.

To show the break-even as a test, assume same bet of $107, but pot is now 2.5 times that amount, or $267.5.

Chop: 80% * ($267.5 * 50% - $107) = $21.4.
Lose: 20% * -$107 = -$21.4.

Yup, that's the break-even.

So a much easier way to do this now that I've checked it is obviously to take the -$21.4 in the losing case and know that we have to earn $21.4 when we chop if we want to break-even. To earn $21.4 when we chop, the pot size needs to be (21.4 / .8 + 107) / .5 = $267.5. That works.

So with all that, I think there are ways to do this quickly at the table with mental math.

First figure out what you lose when you get free-rolled.

It's % chance of losing * bet you're risking.

Here, it's 20% * ~110 ~ call it ~-$22 or so w/mental math. Easy 5 second calc.

Then figure out what you get when you chop.

It's 1/2 final pot adjusted for your bet * % chance of chopping.

The chop is ~300/2 = ~150 - ~110 = 40 * .8 = ~$32.

32 > 22 so call.

Any faster/easier way of doing this intuitively at the table with chops and/or free-rolls?

If anyone reads this and see a math mistake, let me know.

So applying even more rough mental math to the exact situation.. it's like:

Two outcomes. Look at chop, look at loss.

Loss: Gotta call about $110. 20% of the time you lose it. EV ~$-20.

Chop: Calling makes 300 pot. I get 150 in a chop. Of the 150, I already put 110 in. That's 40 net. And that happens 80% of the time. EV ~+$30.

Is there other faster/easier ways to do it?

We have 80% equity in the pot so:

1/2 pot x 80% > call we have +EV

Off hand V over-betting dry pot on a free roll is usually -EV for us.

My method is this: here we have an 80% chance to chop and 20% chance to lose. I just take that 80% and divide it up into a 40% chance to win and 40% chance to lose. Now I go back and add it in to the earlier number. Now we have a 40% chance to win and 60% chance to lose. So if we are getting better than 1.5:1 odds it's a call.

My method could be wrong and if it is someone point it out please. It looks right to me so far though.

Pretty sure that's exactly right.

Probably right. Good one.

My math showed the final pot had to be 2.5x the river bet. Same thing as 1.5:1.