Running it twice with 1 out

This problem has been driving me and my friends crazy.

Imagine a set over set situation, all in on the flop. The lower set only has 1 out to make quads and win. I know that typically, running it twice does not change the EV and only serves to minimize variance. However, in this situation, in which if you win the first run out, you have 0% chance to win the second board, does it behoove the person behind to only run it once? Remember, there is no replacement of the key card.

Proofs would be most welcome.

Thank you.

With only a roughly 4% chance on the flop to win, I'd say they should certainly be begging to run it twice. Maybe even three or four times even though they can only win once, just because something is better than a 96% chance at nothing.

I'd say the player with the higher set should only run it once, or offer to run it four times, guaranteeing themselves 3/4 of the pot.

It doesn't matter. Let's say there are only two players, so there are 44 cards left in the deck once one card is burned and the flop is dealt. To make the math simple, let's say that the burned card is the 4th card of the higher set, so that when the lower set makes quads, it always wins. We also remove any possible backdoor straight and flush draws.

For example: AsAd vs 7h7c. Flop is Ac7s2h. The burn card is the Ah.

The lower set has 4.545% equity in this pot, or exactly 1/22.

What happens if we run the turn and river 22 times?

I agree it has the same EV of running once or twice.

Assume pot is $100 for convenience. Also assume that no board straight or flush possible, so no chop pots.

Run once:

Lower set wins if he gets quads and higher set does not get quads: 2*1/45*43/44 = 4.3434% for EV of $4.34

Run twice:

The first run is same as above but only $50 at stake so EV = 4.3434% * $50 = $2.17

Second run has a few different situations:

A) first run had two blanks, which happens 43/45*42/45 = 91.21% of the time. Then the lower set wins 2*1/43*41/42 = 4.5404% of the time. Total EV contribution of: $50*91.21%*4.5504% = $2.07.

B) first run gave lower set quads so impossible to win second run. Total EV contribution of $0. (Note a small fraction of this scenario had the higher set also making quads, but we dealt with that already in calculating EV of first run).

C) first run gave higher person quads but not lower person, which happens 2*1/45*43/44 = 4.3434% of time. Then lower set wins 2*1/43 = 4.6512% of the time. Total EV contribution of: $50*4.3434%*4.6512% = $0.10

So EV of run twice is $2.17 + $2.07 + $0.10 = $4.34


EV is exactly the same between run once and run twice of $4.34.

Why run anything if not run it once if you are the favorite? I wonder if the underdog would accept it if the favorite offered EV, straight up. Known cards only.

I read question. Took one second to figure out that EV is the same no matter how many times you run it. Obviously the more you run it affects variance, not EV.

Quote: SOOPOO

I read question. Took one second to figure out that EV is the same no matter how many times you run it. Obviously the more you run it affects variance, not EV.

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Sure but OP asked to see the math.

Quote: Mission146

Why run anything if not run it once if you are the favorite? I wonder if the underdog would accept it if the favorite offered EV, straight up. Known cards only.

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They have programs for playing online that will tell you how you are running vs EV. Can be pretty interesting.

Quote: unJon

Quote: Mission146

Why run anything if not run it once if you are the favorite? I wonder if the underdog would accept it if the favorite offered EV, straight up. Known cards only.

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They have programs for playing online that will tell you how you are running vs EV. Can be pretty interesting.

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Yes, and you can also get W/L/T percentage just from the WoV Calculator, so you could use that to calculate your equity in the pot. Rather than run it two, three, four or twenty-seven times, I wonder if anyone would ever agree to spilt based on EV.

Quote: Mission146

Why run anything if not run it once if you are the favorite? I wonder if the underdog would accept it if the favorite offered EV, straight up. Known cards only.

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The value may be in influencing the emotional state of the opponent, not in chips on the table.

It does not! The only thing running it twice (or more) does is to reduce our overall variance while keeping our expected value (EV) intact. In other words, by running it two or more times not only we will make the same amount of money in the long run, but also our bankroll will fluctuate less.

I guess that's my point. As the favorite, my offer would be, "We can either run it once or just chop the pot according to our equity/EV in it."

Sometimes in poker tournaments, the last few remaining players will agree to chop the payouts with a split according to equity each one has ( using the Integrated Chip Model) - rather than play out the rest of the tournament. Mental and physical fatigue are factors in poker tournaments.

Usually someone will not go along with chopping it up. That someone is usually me. I signed up for a tournament with a certain payout schedule
Rarely winner take all but even if that is it. So be it. The kiddie game is down tbe street.

Quote: Mission146

I guess that's my point. As the favorite, my offer would be, "We can either run it once or just chop the pot according to our equity/EV in it."

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Why limit yourself to the book ends of the variance choice?

Quote: unJon

Quote: Mission146

I guess that's my point. As the favorite, my offer would be, "We can either run it once or just chop the pot according to our equity/EV in it."

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Why limit yourself to the book ends of the variance choice?

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I'll happily take the EV chop and be done with it, but I'm giving my opponent the option.