More Math that Doesn’t Matter

In my previous post, I described a big decision – either fold or commit $500+ to a likely 3-way all-in on the turn – where the Expected Value was very close to zero.

Now I’ll consider another possibility. Let’s reset the scenario:

I have J8, and the board is Q92 – 6♠. After flopping a gutshot straight flush draw, the turn was a total miss, both for me and for the villain’s most likely ranges. There is $615 in the pot already, and I have $510 remaining.

The villain on my right was the big blind, and had 3-bet to $50 pre-flop, then C-bet and called my raise on the flop. For purposes of this blog, I’ll call him “Omar.” Omar’s range is very narrow — AA [spoiler: #TheyAlwaysHaveIt] or KK or AK, for a total of 13 combinations. If he had QQ and flopped top set, after my raise and another player’s call on a drawy flop, I would expect him to re-raise right away. He would not 3-bet pre-flop with Q9, 99 or 22. With anything weaker than QQ, he would simply fold to my raise on the flop. After the turn card was dealt, Omar checked to me. He has me covered.

The villain on my left had called my pre-flop raise to $20, then called again after Omar’s 3-bet and my call. For purposes of this blog, I’ll call him Billy. Billy wanted to re-raise all-in on the flop, but in his excitement he declared “I call all-in” before pushing out his chips. The dealer ruled that as soon as the first two words “I call…” were said, he was precluded from raising. I have to assume Billy likes his hand, so his range will include top pair / top kicker, nut flush draws, open-ended straight draws, and 2-pair plus types of made hands. Given that he called but didn’t raise twice pre-flop, I can eliminate over pairs, top set, top + bottom pair, and bottom 2-pair from the made hands. I’m not sure how he would play Q9o or JTo, but he could have Q9s or JTs. There are eight possible nut flush combos after eliminating the diamonds on the board and in my hand. Would he actually call twice pre-flop with the weakest suited aces? The strongest K-high flush combo he can have is KT and I can eliminate any weaker flush draw hands. If he does have a flush draw it will be stronger than my flush draw, although both of my cards would be live against him as would my straight draw.

That leaves us with a range of AQ (12 combos), A♦2♦+ (8 combos), K♦T♦ (1 combo), Q9s (3 combos), and JTs (3 combos)… a total of 27 combinations. Notice how different Billy’s range is from Omar’s. He has $430 remaining.

One of the interesting dynamics is Billy’s clumsy and failed attempt to go all-in on the flop followed by Omar’s over call with full knowledge that Billy wanted to ship it. Maybe I can use that to my advantage???

After Omar checks on the turn, I can beat Billy to the punch and go all-in myself. Since I’m not folding regardless, perhaps a shove from me will cause him to re-think his situation with the weaker part of his range. [Spoiler: from the previous post, we already know he had AQo, for top pair / top kicker.] Can I make him fold? If he does, will Omar read the memo and fold too? Is it wise to try?

How does the math work? [So glad you asked.] In the previous post, I showed how my projected final stack would be $531 after a 3-way all-in, making the EV of that play +$31.

If both villains fold, my final stack will be ($510 + $615 = $1,125). Let’s assume there’s a 10% chance of pulling that off, and the other 90% of the time they both call. I’ll win $2,065 another 23.6% of the time (my equity of 26.2% x 90%), and go busto the remaining 66.4%. My projected final stack is now $600, making the EV of this shoving + $90.

Here is the math:

Both fold –> 10% x $1,125 = $112.50

Both call, I win –> 90% x 26.2% x $2,065 = $487.34

Both call, I lose –> 90% x 73.8% x $0 = $0.00

($112.50 + $487.34 + $0.00 = $599.84), which rounds to $600.

If they both fold 20% of the time, my projected final stack increases to $658, making the EV of shoving + $148. While it didn’t work this time, if there is any chance at all that it might, being the first to shove is superior to checking to let Billy shove, then calling after Omar calls.

It’s also possible that Billy would check back on the turn if I checked (in which case I would retain my stack), or that one of them would call my shove but the other would fold (leaving me the same number of outs to win, but a smaller reward if that happens). Those will have to be math problems for another day.

Oh, and another thing: if both of these guys and others like them are shoving or calling off stacks of this size with one pair, I should never, ever, ever, ever, ever, ever bluff again.

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