fennaixelphox | Poker thoughts regarding a deck containing one joker. (Reply)

So, I've been thinking about poker lately, and the probability of each hand in a regular deck vs. with one joker in the deck. The probability of each hand, given 0 or 1 jokers in the deck, is as follows. ¹

Each hand type is ranked according to the number of unique hands you can make of that type. The lower the number is, the more rare that hand is, and thus, the better it ranks.

Hand Type	With 0 Jokers	With 1 Joker
Five of a Kind	(impossible)	13
Straight Flush	40	204
Four of a Kind	624	3,120
Full House	3,744	6,552
Flush	5,108	7,804
Straight	10,200	20,532
Three of a Kind	54,912	137,280
Two Pairs	123,552	123,552
Pair	1,098,240	1,268,088
High Card	1,302,540	1,302,540

It is impossible to make a High Card hand with any selection of cards that includes at least one joker, because the joker can always be matched to the highest ranking card in your hand.

You might have noticed that the number of Three of a Kind hands with one joker is actually MORE than the number of Two Pair hands with one joker. Part of this has to do with how any hand of the form XXJYZ (two matching cards X, one joker J, and two unmatching cards Y and Z) can form either a Three of a Kind (if the joker is treated as an X) or a Two Pair (if the joker is treated as a Y; we'll assume that the unmatching cards are always sorted by rank, so the joker will never be treated as a Z because to do so would make a lower hand). To be exact, there are 82,368 possible hands of the form XXJYZ. However, if we're sorting the hands by how rare they are, that should mean that Two Pair outranks Three of a Kind, right? Let's take a look.

Hand Type	With 0 Jokers	With 1 Joker
Two Pair	123,552	205,920
Three of a Kind	54,912	54.912

Oops! Once again, the more common hand outranks the less common one, because the 82,368 hands that would have formed a Three of a Kind are now used to form a Two Pair. This won't do at all!

So how do we fix it?

The way suggested in the Data Genetics article that made me think of this is taken from Pai Gow Poker. In that version, a joker can only be used to complete a Straight, Flush or Straight Flush. In any other hand, it is treated as a suitless Ace. (That means that, for instance, a hand of 7777J is a Four of a Kind, not a Five of a Kind, because the Joker would become an Ace instead of another 7. The only true Five of a Kind, then, would be AAAAJ.) The term for a joker that's limited like this is a "bug", and it changes the hand rankings as follows: ²

Hand Type	With 0 Jokers	With 1 "Bug" Joker
Five of a Kind	(impossible)	1³
Straight Flush	40	204
Four of a Kind	624	3,120
Full House	3,744	6,552
Flush	5,108	7,804
Straight	10,200	20,532
Three of a Kind	54,912	63,360
Two Pair	123,552	138,600
Pair	1,098,240	1,215,024
High Card	1,302,540	1,418,964

These hand rankings work perfectly fine, and the "proper" ordering of hands can be preserved without needing to resort to resolving a paradox, but what if you're like me and you want the joker card to be truly wild?

That got me thinking, why does XXJYZ have to be categorized as EITHER kind of hand? Its probability is greater than that of a Three of a Kind, but less than that of Two Pair. It is this thought that made me consider turning XXJYZ into its own hand type, which I have tentatively named the "Troupe."⁴

Hand Type	With 0 Jokers	With 1 Joker
Five of a Kind	(impossible)	13
Straight Flush	40	204
Four of a Kind	624	3,120
Full House	3,744	6,552
Flush	5,108	7,804
Straight	10,200	20,532
Three of a Kind	54,912	54,912
Troupe	(impossible)	82,368
Two Pairs	123,552	123,552
Pair	1,098,240	1,268,088
High Card	1,302,540	1,302,540