Note: Exclusive to the new Poker1 site.
The practice of “running it twice” (or even more times) has become a common practice in some poker games. What it usually means is that, when a heads-up hand allows no more betting, because one player is all-in, the opponents can agree to see the outcome dealt twice, instead of once.
A player winning both times claims the entire pot. Otherwise, the pot is split.
Should you agree to run it twice? Does it matter whether you’re the favorite or the underdog? Here’s a technical explanation, with the answer at the bottom.
A $200 pot
Imagine that it’s a $200 pot. How the money got there doesn’t change the decision. Also imagine that you’re a 3-to-1 favorite to win. If you only see one outcome, you’ll win three-quarters of time, assuming you replay the result forever. That means you’ll have a gross win of $200 three times, on average, for a $600 total win and lose $200 once. The advantage is $400 every four pots or $100 per pot.
For a single pot the math is (.75 times $200 = $150) – (.25 times $100 = $50) or $100.
If you play it twice for each of the four instances, you’re likelihood of winning both “runs” is .75 times .75, or .5625. So, 56.25 percent of the time, you’ll take $200. You’re likelihood of losing both hands is .25 times .25 or .0625. So, 6.25 percent of the time you’ll take nothing.
More math
Those two possibilities account for 62.5 percent of the outcomes, when added together. The remaining 37.5 percent of the time, you’ll split the pot, meaning no gain either way. So, the math is (.5625 times $200 = $112.50) – (.0625 times $200 = $12.50) per two-run events. That’s still a $100 advantage per hand, because on the remaining 37.5 percent, you split and take $100.
The full math for a double run would be:
((.5625×200)+(.375×100))-((.0625×200)+(.375×100)) = 100
The simple answer
You can see that it doesn’t make any difference in the long run whether you take the deal or not. And it doesn’t matter to the underdog, either. The process reduces the risk and makes it more likely that the outcome will be closer to expectation. But that’s all it does. — MC
If you run it twice isn’t your odds of winning the second run less than 0.75? (Assuming you won the first and the villain would have same number of outs but there are 2 less cards in the remaining deck?
Whether or not cards are shuffled between runs, the odds don’t change overall. Without a shuffle, the first run cards certainly do affect the odds for the second run, but so what? The cards aren’t known unless you look at them, so their influence is mathematically unknown before the run-twice process begins. Suppose there were just two cards left in the deck, an ace and a king. If you only deal one card, the chance of an ace is 50 percent. If you deal both cards, that 50 percent is absolute, regardless of whether the ace comes first or second.