You have AA on a board of: JT7A

Your V goes AI & it's folded around to you. You know this V extremely well & know that he would never, ever, go AI without Broadway. It's not happening, ever. He never drinks, never tilts, never does anything other than ABC. You know he has Broadway. His cards might as well be turned face up. No exceptions.

This is a math problem, not a "put your V on a range" situation.

The pot, before V's AI, is $725 & you have put $150 in the pot.


Your V goes AI for $320, giving you $1045:$320 odds, or 3.265:1 on your money, as a 3.405:1 dog.

So, 3.405x you lose $320 = -$1089.60
1 time you win $1045 - $1089.60 = $44.60 -Ev long term.

However!

Before calling the AI, you had already put $150 into the pot. So, aren't my choices: 1. Fold to the AI & lose $150 long term, or 2. Call & lose $44.60 long term?

Having a hard time following the action:

If you're doing this live, you can calculate that you need about 25% (23.5%) equity to call if you're calling $320 to win $1365. Your equity vs. a 100% range of KQ (if he showed it to you face-up) isn't enough to call. If his range isn't 100% KQ then it's a call. I'm not sure what you're asking here? It's pretty clear-cut. This is too easy of an example because it's a call every time, provided your V doesn't show you his hand first.

If you're saying you put $150 in and and then he put $170 more in on top, this becomes even easier. You would only need 12.5% equity to call. You call every time even if he shows KQ face-up. It's not -EV in that case, it's +EV. You don't lose long-term. Your math must be off somewhere if you're saying calling $170 to win $1365 with 20%+ equity is -EV.

No.

Your choices are to lose $0 *more* or to lose $44 *more*.

The $150 is gone already no matter what choice you make. It now belongs to the pot.

What you put into the pot prior to the all in is irrelevant. Once it leaves your stack, it belongs to the pot, not you.

Math is you need 23% equity to draw to make your total EV = $0 (meaning that calling and folding have the EXACT same expected Value). So, if V turns over KQ, then we know we have 10 outs out of what is now 44 unseen cards. 10 we win, 34 we lose, so 22.7 % of the time we win and 77.2% of the time we lose.

So if we choose to call anyway, we are making a mistake of 0.3% * $320, so calling costs us $0.96 in the long run.

In practice however, we can never know with absolute certainty that V only has KQ so clearly this is an auto call. I would also argue that depending on if a given call that is slightly -EV (say less than 5%) will give us an image that is "crazy/gambly", then we likely can make up that lost equity by calling anyway even when we lose.

Dammit iraisetoomuch! You spoiled all the fun!

I also counted the $150 I had already put in the pot as part of the profit the 1 time I make my boat on the river.

When I read his post a few more times it looks like he's saying you bet $150 and then V shoved $320, which makes it a +EV call even if he shows you KQ face-up.

This is the sunk cost fallacy. The amount of money you put in the pot already doesn't matter, just your pot odds and equity.

If you count folding as -$150 long term, then you should actually count calling as -$194.60 long term. When you say folding is -$150 long term, you're referencing the beginning of the hand. When you calculated that calling is -$44.60, you were referencing the current decision point of call/fold. You have to use the same reference point in a comparison.

Also, you calculated the EV of calling incorrectly, and were off slightly on the odds of winning the hand. We are exactly 3.4:1 against, with 34 cards that lose and 10 that win, not 3.405.

Your EV of calling is incorrect because you tried to do a weighted average without dividing by the sum of the weights. The correct way is:

(3.4*-320+1*1045)/4.4 = -$9.77

I read it differently, like, of the $725 in there before V shoves, we have contributed $150.

So we are getting $1045 to call $170 more ($320-$150)?? Ummm, as Mike Sexton used to say......."Vince, he beat him into the pot."

Why is THAT a thread?

+1

At this point it's irrelevant how you put money into the pot / how much you put into the pot (for better or worse). All you can do at this point is make the best decision at this point.

ETA: As for the actual decision itself, the OP is a little confusing but I'm simply going to assume we're being asked to call $320 to win $1045. I always the Rule of 2 and 4 shortcut to estimate my equity at the table, so if he has the straight I have 10 outs so about 20% equity so need about 4:1, and I'm coming up slightly short of that getting just over 3:1. But unless our opponent has literally turned his straight up on the table, I'm simply going to assume that he can sometimes (although not always) do this with worse sets (at the very least), so close enough for me to probably not be able to fold.

GcluelessNLnoobG

I'm no math wiz by any means, so someone please correct me if I'm wrong, but I always divide the amount I have to call by the entire pot size(including my call) to figure out the amount of equity I need to break even on the call.

In this case, it's $320 to us and the total pot after my call will be $1365. So 320/1365 is somewhere around 23%. With a set vs only the nut straight here on the turn we have right around 23% equity, so it's pretty much a shrug call even if he shows us his cards.

Am I off here at all?

Yeah, but in that case it would mean the pot is $725 including your bet on the table of $150 already in front of you, so in that case you're putting in $170 for $1215 which is 14% which is still +EV, just not as +EV as if the $725 wasn't already including the $150. It's still a situation where you beat him into the pot.

The only situation where you fold to KQ face-up based on the original example is if the pot is $725 and V shoves $320 and you have to call $320.

Nope your good

Your math is right but your hand equity is closer to 20% and not as much as the 23.5% pot odds in your example, so it's a slight fold.

We're still trying to clarify what he meant in the OP but in that situation you described, that's the only situation where you fold to a face-up KQ. You'd be better off putting your $170 on the passline on a craps table.

This is the correct way to look at things regarding equity. I come from Limit so I typically look at things in terms of odds, but whatever floats your boat.

If Villain *only* shows up with straights here, then we're not quite getting the odds (or equity or however you want to look at it) to make the call as we're falling just a little bit short (my Rule of 2 and 4 estimate gets us only to 20% although PokerStoving it gets us a little closer but still technically short). Still, I sigh call due to most villain show up with worse a small percentage of the time (at least).

My guess is that OPs question was more to due with how to react to how much he's already put into the pot (where the answer is to totally ignore that).

GcluelessmathnoobG

You need to use more than two digits of accuracy. This is a very close spot.

Also, in a real scenario you can not be certain his range is 100% KQ. It is such a close spot that even if he has something else a very small percentage of the time it would become a call. In practice I would call even if I think villain never does this without KQ, because I can't really be 100% sure.

OP, V goes AI for $320 more, or $320 total ($170 more)?

Big difference.

Quick math I do at the table in these spots. 10 outs once to make a boat or quads, and there are 44 unknown cards remaining (52 - 2 in my hand - 4 on the board - Vs KQ). So I hit my hand 1 out of 4.4 , making me a 3.4 to 1 dog. So in game, I know I need 3.5 to 1 or better for the call to be profitable.

In game it wouldn't take me but a few seconds to realize this is a slightly losing call against the nuts, but I hate folding top set (or sets in general really) and would tell myself that the price is close enough and V won't have KQ 100% of the time (even though he may very well) , and then call it off.

Tbh. If you never fold top set you're probably ok.

*GRUNCH*

This has probably already been answered but I'll give it a go. My degree is in mathematics which I hope lends some confidence this explanation and calculation are correct.

The 150 is part of the pot already. It's gone.

You did not account for the 150 when you win.

Your pot odds are also wrong. The pot odds are (725+150+320):320 or 1195:320 or 3.734:1

You then want to convert the pot odds to a probability. If the pot odds are x:1 then you need to win 1/(x+1) or more.

Here x is 3.734 so you need to win at least 1/4.734 = .211

You have 10 outs vs Broadway. Rule of 2 gives a rough estimate but the actual probability you win is a bit higher.

P(win) = 10/(cards not seen)

Normally OTT cards not seen = 52 - 6 = 46

However if we are sure he has KQ then cards not seen = 44

So P(win) = 10/44 = .227

Since .227 > .211 you have odds to call

If you have any follow up questions please let me know.

Okay, rereading I'm not clear on whether the pot was 725 before you put in 150 or after. If it's before, my above post is correct. If it's after we need to calculate the pot odds again which would be

(725+320)320-150) or 945:170 or 5.559:1

Converting to a probability we would then need to win 1/(5.559+1) = 1/6.559 = .152

We still have P(win) = .227

In this case it's a snap call since .152 << .227

Note it also isn't totally clear if villain goes all in for 320 or for 495. I'm assuming 320. Please correct if this is wrong as it changes the calculation. And please clarify as to the exact action.

All these results could be wrong if I'm reading action incorrectly so I'll outline how to compute this generally.

In general let's suppose the pot is P after villain's bet. It costs us C to call. The pot odds are then P:C and conversion to a probability is C/(P+C)

For example, pot is 400. We bet 200 and villain shoves for 750 total or 550 more. The pot P is 400+200+550 = 1150. It costs 550 to call so C = 550.

Pot odds are 1150:550 or 2.0909:1

Converting to a probability we can use 550/(1150+550) = .3235 which note is the same as 1/3.0909

We then compare this number to P(win). If P(win) > C/(P+C) we call

Damn! I thought for sure I was clear, but as Shai points out, I wasn't.

I meant that Hero had put in $150 on the previous street, so my math is correct.

Hero was last to act & V went all in OTT for $320

There is $1045 in the pot [725+320] / your $320 call = 3.265:1 on your money on a 4.05:1 draw.

This is NOT a real life situation. You can assume V turned over his hand to try & get you to fold. I doubt anyone folds here in real life with top set not knowing their V extremely well.

I understood your post and explained why your math is not correct.

Okay in that case your pot odds are correct but this is actually a 34:10 or 3.4:1 draw since 34 cards leave KQ with the nuts and 10 cards (1 A, 3 J, 3 T, 3 7) give you a full house or better.

Since 3.265 < 3.4 it's a fold but barely.

Clear now. Here's how I math it at the table.

If I call, my $320 will represent a little under 25% of the $1365 pot, so I need ~23% equity to call (actual number is 23.44%, but I wouldn't take the time for long division at the table).

I have 10 outs to boat/quad up. By rule of 2 and 4, I have about 20% chance to hit. Normally I stop there, since it is so close that if he has one combo that's not the nuts, I know I'm getting a good price to call.

Let's say he actually showed us the nuts to try to get us to fold, though. In that case, we know his 2 cards, so as Shai said there are only 44 unknown. In that case I estimate that 10 is a bit more than 22.5% of 44 (it's actually 22.73, but again, I don't do the long division at the table).

Now I know that it's so close that I really ought to break out the division, but I also know that it makes little difference either way and I've already been tanking for a while. Since this guy is such a nit as to show me that nuts to try to get me to fold, I figure that if I'm a few bucks short in actual equity, I more than make up for it in tilt equity.

So I call and then do the actual math later to figure the fine grain of whether I was technically +EV or not (spoiler alert: not).

While strictly in a vacuum, it's a slight fold, if there are other players on the table with large stacks that have you covered, it could tilt a slight fold into being profitable in the long run of your session when you consider implied winnings on future hands by having as deep or a deeper stack than players you may consider inferior. The added value of tilt could be +EV as well, as Garick noted. This obviously isn't worthwhile when you're talking about it being -EV by a non-nominal amount, but if you're talking <1% it might be worth consideration.