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Tough math/theory question about card removal

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Posted by posted in Gen. Poker

Tough math/theory question about card removal

Hey guys,

I was curious how much your opening range from the SB should differ in a 9-handed game compared to shorthanded games. Basically, considering that when this situation occurs in a 9-handed PLO game, over half the deck is already folded away and the mucked cards will be very strongly weighted towards low cards.

In other words, when you open the SB here in a 9-handed PLO game, I suspect the chance that the BB has a top 15% hand is SIGNIFICANTLY higher than 15%... so that implies our SB opening range in a 9-handed game should differ than our SB opening range in a 3-handed game, however I've never seen any literature on this topic before.

Thoughts?

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If the above question is too complicated to solve which I suspect it may be, here's a simplified version which might serve as a decent estimate:

No one will ever fold any hand with at least two Aces, and everyone will fold a hand with one Ace X% (let's say X=75%?) of the time. If it folds to you in the SB and you have no Aces in a 9-handed PLO game, what are the chances BB has at least two Aces?

I really wish I could solve this myself but I suck at math. Thanks!

5 Comments

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ZenFish 11 years, 11 months ago
I have no source for this, but I remember some estimates of this "bunching effect" done for Hold'em. The conclusion was that it was negligible.

When the first 7 players fold, they fold a lot of low cards, but most of the hands they fold are raggedy hands with both high and low cards in them. K953 and such. So a lot of high cards are folded away as well.
jonna102 11 years, 11 months ago
Two high cards and two low cards with one suit will be your average PLO hand. Most of these hands are garbage, so they are folded. But it means that the number of folded high and low cards should be (on average) roughly the same. You could do some calculations on this, but I don't think it's really necessary. If the effect is there it's likely to be small. If the effect was significant I think players would pick up on it, and I've not noticed (or heard of) anything like that myself.
GameTheory 11 years, 11 months ago
Ok if it was HU and you don't hold an ace in the SB, the chance of the BB having at least two aces is:
5853/194580 = 3.0%

Now 9 handed PLO. We assume the first 7 players play {AAxx,KKxx,AKxx} only, this is a 10.7% range for each player so reasonably tight.
Now I ran a simulation that shuffled 2,000,000 decks.
523,824 times it was folded to us on the SB while we held no ace. 24,082 of those 523,824 times the BB has at least two aces, or a conditional probability of 4.6%.
This means that the chance the other player has two aces is increased by more than 50%.


If we use wider ranges for the first two players, {AAxx,KKxx,QQxx,AKxx,AQxx,KQxx}, or a 22% range for each player, it gets folded to us 136,248 times on the SB while we hold no ace. 7,805 of those 136,248 times he BB has at least two aces, or a condional probability of 5.7%.
This is an almost doubled probability.

This looks a little bit more counterintuitive, but it is very logical. The constraint that the SB holds no aces and the out of the first 7 players no player holds two cards ace, king or queen is very strong.
If there are two aces dealt to the first 7 players, then the probability that either of them also holds another ace, king or queen becomes very high. The times that this does not happen coincides with the times that the BB holds a lot of aces.
Aesah 11 years, 11 months ago
Thanks, very nice post!

So based on your simulations, would you expect it to matter in practice (i.e., should an SB opening range differ in a 3-handed and 9-handed game?) Since of course many AKxx hands are trash, many lower rundowns are good, etc., but in general high card hands are still more likely to be played.
GameTheory 11 years, 11 months ago
Yes I expect it to matter. The looser the games are, the less likely nobody of the first 7 players has a hand. So when they do all fold they must've had a lot of trash. Hence it is much more likely that the BB has good cards. Having A(K/Q) is having 8 kickers, which is similar to Ax suited with a connector or something.

And what you say is obviously true, a lot of playable hands like T987 don't have blockers, so the effect is obviously smaller than 3.0% => 5.7%.

I would estimate that 4% is a realistic estimate in most typical games.

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