For those who don't speak Mathhead, in probability, binomial means "with replacement" (i.e., you're reshuffling the used cards back into the stub between runouts) and hypergeometric means "without replacement" (i.e., the way RIT works in real life). The EV for both players is unchanged in both cases.

Using my 36:8 river example from earlier, the way I analyzed it was the normal way, without replacement (hypergeoemtric). Running it exactly twice:

For those who wanted some kind of mathematical evidence that running it twice is EV-neutral, this is one such data point. That 81.8% is the same approximate probability of Hero winning on a single runout (36/44 ≈ 81.8%), and it comes out exactly the same if you use the precise values instead of rounding. If anyone really wants me to strip out the specific values and algebraically prove the general case, I can do that too, but the explicit numbers are usually more helpful for non-math folks.

Switching between the two models does not change equity (and thus EV) at all. The only real effect is that Hero has a slightly greater chance of scooping and a slightly lower chance of splitting. On the other side, Villain has a slightly greater chance of splitting and a slightly lower chance of scooping. The favorite is more likely to scoop and less likely to split, and the underdog is less likely to scoop and more likely to split, but it all comes out in the wash.

The only difference is in variance, which is greater when you run it out multiple times with replacement than without. To see why informally, look at the extremes. With replacement, it becomes possible, though unlikely, for either player to win 44 consecutive runouts. Without replacement, it's fully impossible for Villain to win more than 8 out of 44 runouts (or for Hero to win more than 36 out of 44).

I have a math degree. This is all good and everything. But why the hell are we getting into this in this thread? Is leaving it at "running it multiple times doesn't change the EV," not enough?

For those who don't speak Mathhead, in probability, binomial means "with replacement" (i.e., you're reshuffling the used cards back into the stub between runouts) and hypergeometric means "without replacement" (i.e., the way RIT works in real life). The EV for both players is unchanged in both cases.

Using my 36:8 river example from earlier, the way I analyzed it was the normal way, without replacement (hypergeoemtric). Running it exactly twice:

For those who wanted some kind of mathematical evidence that running it twice is EV-neutral, this is one such data point. That 81.8% is the same approximate probability of Hero winning on a single runout (36/44 ≈ 81.8%), and it comes out exactly the same if you use the precise values instead of rounding. If anyone really wants me to strip out the specific values and algebraically prove the general case, I can do that too, but the explicit numbers are usually more helpful for non-math folks.

Switching between the two models does not change equity (and thus EV) at all. The only real effect is that Hero has a slightly greater chance of scooping and a slightly lower chance of splitting. On the other side, Villain has a slightly greater chance of splitting and a slightly lower chance of scooping. The favorite is more likely to scoop and less likely to split, and the underdog is less likely to scoop and more likely to split, but it all comes out in the wash.

The only difference is in variance, which is greater when you run it out multiple times with replacement than without. To see why informally, look at the extremes. With replacement, it becomes possible, though unlikely, for either player to win 44 consecutive runouts. Without replacement, it's fully impossible for Villain to win more than 8 out of 44 runouts (or for Hero to win more than 36 out of 44).

I have a math degree. This is all good and everything. But why the hell are we getting into this in this thread? Is leaving it at "running it multiple times doesn't change the EV," not enough?

I have a math degree. This is all good and everything. But why the hell are we getting into this in this thread? Is leaving it at "running it multiple times doesn't change the EV," not enough?

My last post was because @Rakrul had inquired about whether anything changes between the with/without replacement approaches. I had to indulge. Topics like this (i.e., analyzing optional bets that may or may not be fair to all participants) are interesting to me from a gamesmanship perspective. Hell, I published a whole book analyzing the effect of straddling on EV under different circumstances.

My last post was because @Rakrul had inquired about whether anything changes between the with/without replacement approaches. I had to indulge. Topics like this (i.e., analyzing optional bets that may or may not be fair to all participants) are interesting to me from a gamesmanship perspective. Hell, I published a whole book analyzing the effect of straddling on EV under different circumstances.

I'm willing to bet if you guys somehow had gotten it all in on the flop, he would have turned his hand over and declared "open ended" or "double gutter." You know... 4679! Oh... Wait...

Flopped quad kings but high hand was already a straight flush and 15 minutes left on clock

Next hand get it in with double suited AA64 and lose to JJTT when he flops a Jack

Shortly after have AAK2 double suited in a raised multiwaypot and see a J86 two of my suit flop, plus nut backdoor suit. Get it in 3 ways and whiff, up against JJxx again