October 30, 2009

PLO: Exactly 6 times as sick as NLHE

In my last post I showed how to figure out what % of a player's range certain hands make up in PLO and how that compares to NLHE. An observant reader might have noticed that the answer for PLO was always 6 times the answer for Hold 'Em. Coincidence? No.


Consider a single two card combination, like AsKs, just to take an especially juicy two cards. In Hold 'Em, there's obviously exactly one hand that contains those two cards. In PLO, there are 50*49/2 = 1225, which is the number of combinations of any two cards that aren't the As or the Ks. So there are 1225 times as many hands that have AsKs in PLO as in Hold 'Em. How many total hands are there in PLO vs. Hold 'Em? In Hold 'Em there are 52*51/2 = 1,326. In PLO there are 52*51*50*49/(4*3*2) = 270,725, or 204 1/6 times as many. And 1225/204.1666 = 6. So a given two-card combo is six time as likely to be in an Omaha hand as a Hold 'Em hand. So while AsKs is 1/1,326 = .0754% of all Hold 'Em hands, the Omaha hands including AsKs are 1225/270,725 = .4525% of all Omaha hands.

This might not be too surprising since each Omaha hand includes six possible Hold 'Em hands (first card/second card, first/third, first/fourth, second/third, second/fourth, third/fourth). However, they are not independent. So it's not just like dealing a hold 'em hand, reshuffling the deck, dealing another hold 'em hand, and repeating six times and seeing if any of the hands dealt were AsKs. This would be (1-.000754)^6 = .4516%. On the other hand if you dealt out six Hold 'Em hands, without putting the already dealt cards back in the deck before dealing the second, third, and later hands, the odds of dealing AsKs are .5629%. It's not totally clear whether an Omaha hand should be more or less likely to have AsKs. The fact that the first and second cards are not the As and the Ks makes it MORE likely that the third and fourth will be. On the other hand, it makes it LESS likely that the first and third cards, for instance, will be. In any case, it turns out that the effects kind of balance out, so that the final result is that AsKs is included in .4525% of all Omaha hands, much closer to dealing 6 independent Hold 'Em hands with card replacement than dealing them without replacement.

Oh, and just so this post isn't totally useless and boring, if you've made it this far, here's a 900 bb pot I won vs. a complete monkey who had absolutely no fold button:


-BRUECHIPS

5 comments:

The Poker Meister said...

25PLO? 900 BBs deep? I'm not getting it; is this typical or was this guy a major reload fish?

spritpot said...

You mean why am I playing 25PL or why did he play his hand like that? We're actually just a little over 400 bb deep, then the other guy contributed another 100 bb to the pot.

The Poker Meister said...

How do you get 400 bb deep is the question.

spritpot said...

Ah how did that guy run up a big stack? He was up to 10 bi at one pt, in no small part due to me trying to bluff him a cpl of times. If you're playing super LAG in PLO, which this guy obv was, there's a decent chance you will run up a big stack, no matter how bad you are. It's not like in NLHE where the donkeys can get it in w <20 % equity all the time with there tpnk vs tptk or set. If you're getting it in w > 65% equity in PLO, that's a monster edge. Even in this hand, he's got 25% equity or something 3-way, so he could easily outflop me and win the monsterpotten himself.

The Poker Meister said...

I'm never playing PLO.
:-)