PLO Math: Answers

Ok well, here goes:

First, the Hold'em warmup: there are 3 combinations of 66. You have two cards in your hand, there are three on the flop, so there are 47 cards unaccounted for. Since he was calling with any hand preflop, his entire range is all two card combinations of those 47 cards, which is 47*46/2 = 1081. So 66 makes up 3/1081 ~ 0.3% of his range. There are also 3 combinations of 33 and 99, so 9 combinations of sets, so sets make up ~ 0.9% of his range. If he calls with only 30% of hands preflop, then this increases, because we're taking a bunch of junk out of his range, so the number of combos he could possibly have is ~.3*1080 (*), so then 66 makes up ~ 0.9% of his range, and all sets make up ~2.7%.

Now that we see how to do the two-card problem, let's move on to the four-card (Omaha) problem. There are still 3 combinations of 66. But there are also two other cards in the player's hand. There are 43 cards left in the deck (52 - your 4 cards - 3 frop cards - the two sixes), so 43*42/2 = 903 ways to arrange the other two cards in his hand, for a total of 3*903 = 2709 hand combinations that include 66. There is a total of 45!/(41!*4!) = 148,995 ways to choose 4 cards out of the remaining 45 in the deck. So hands including 66 make up 2709/148,995 = 1.8% of the villain's range. Much more than the 0.3% from Hold 'Em.

Now consider all hands that include at least one set. We could just do 3*2709, and that's almost right, but it double counts TT66, 6633, and TT33. There are 9 combinations of each of those hands. So the actual number of combination of hands that include at least one set is 3*2709 - 3*9 = 8100 combos. So hands including at least one set make up 8,100/148,995 = 5.4% of all hands.

Just as before, if he calls with only 30% of hands preflop, then there are only ~ 148,995*0.3 = 44,698 combos possible (*). Then 66xx makes up 6.1% of his range, and he has at least one set 18.1% of the time. Sick!

Moral of the story: There's a good chance you will run into some very good hands in Omaha. Whereas in Hold 'Em when someone is repping a set and a set only, usually they are bluffing, if only because they only flop sets less than 3% of the time. In PLO, however, opponents make sets often enough that they would have to be raising a TON of flops for them to be bluffing a very high % of the time.

-BRUECHIPS

(*) - In order to do this, we have assumed that the cards on the board remove just as much of his pre-flop calling range as from his pre-flop folding range. This will never be exactly the case, but it's pretty close, and figuring out the exact number is ridiculously complicated and involves specifying his entire calling range, so it's way more trouble than it's worth.

Another note: 32% of PLO hands contain a pair, so if the villain is calling with any TT, 66,

and 33, certainly he is calling more than 30% of hands preflop.