Poker Puzzle

You and some buddies are playing dealer's choice poker. One time the dealer chooses draw poker, 2 cards to be drawn, deuces wild. (Alright, wimpy game, but he picked it.) One of the players asks if the order of winning hands is the same as usual, from royal flush on down. He says, "Like I'm thinkin' 'bout three of a kind against two pairs." You, being an MIT-trained statistician, tell him the order of winning hands depends on their predicted relative frequency of occurrence. He says, "Yeah, that's what I'm thinkin' 'bout." What, if anything, is the basis of his concern? How is it resolved?

Originally posted by JS357
You and some buddies are playing dealer's choice poker. One time the dealer chooses draw poker, 2 cards to be drawn, deuces wild. (Alright, wimpy game, but he picked it.) One of the players asks if the order of winning hands is the same as usual, from royal flush on down. He says, "Like I'm thinkin' 'bout three of a kind against two pairs." You, being an MIT- ...[text shortened]... thinkin' 'bout." What, if anything, is the basis of his concern? How is it resolved?

I think you would be talking about the wild cards, the number of them in a deck, 2, I think, would screw up the statistics of the hand expectancy vs not using wild cards.

I have now idea what that would be but it looks like that would add another level of statistics to be dealt with.

Actually, thinking about wild cards, in dealer game, the dealer also specifies what is wild, one eyed jacks, and the like, 2's, whatever the dealer specifies, that would certainly add another level of statistics.

Originally posted by sonhouse
I think you would be talking about the wild cards, the number of them in a deck, 2, I think, would screw up the statistics of the hand expectancy vs not using wild cards.

I have now idea what that would be but it looks like that would add another level of statistics to be dealt with.

Actually, thinking about wild cards, in dealer game, the dealer als ...[text shortened]... like, 2's, whatever the dealer specifies, that would certainly add another level of statistics.

OK, that's the basic reason, and here's a detailed answer.

Everyone who has two pair, say AAJ52, using one wild card deuce, also has three of a kind, and will choose that if it beats two pair as it does in no-wild poker. But that makes two-pair rarer, because only natural two-pairs will be played as two pair. But if you rank two pair as beating 3 of a kind because it is more rarely played, everyone will choose two pair over three of a kind with a BBCD2 hand, and the reversal reverses again. So with deuces wild there is no rational odds-based basis for ranking the hands. The odds reversal also affects 4 of a kind.

This table is from: http://www.play-poker-card-games.poker.tj/probabilities-holding.html

I hope this table prints OK. The page before it, Using Wild Cards, is also relevant.


Probability without wild cards Probability with deuces wild

One pair 1 in 2.4 1 in 2.1

Two pairs 1 in 21 1 in 27

Three of a kind 1 in 47 1 in 7

Straight 1 in 255 1 in 39

Flush 1 in 509 1 in 197

Full house 1 in 694 1 in 205

Four of a kind 1 in 4,165 1 in 84

Straight flush 1 in 69,974 1 in 638

Five of a kind n/a 1 in 3868

Royal flush 1 in 69,974 1 in 5370