Calculating Probability with a Wild Card

[JoeU]

There is a home game in my area that uses a "wild card" in the game. The people who play the game state that a 5 of a kind now ranks ahead of a straight flush or royal flush. Is this mathematically correct? You could technically have a full house become a 5 of a kind, but this hand, by no stretch of the imagination, beats a straight flush. In this instance, there are 8 cards in the deck that make a 5 of a kind, and 9 cards that make a straight flush. But on the other hand, in a normal game, there are 5 cards that make a straight flush, but only 4 cards that make quads. I'm not a math wizard, so I was hoping that someone here could answer these questions.

Thanks
Joe

Dear Joe,

you wrote:
"There is a home game in my area that uses a "wild card" in the game. The people who play the game state that a 5 of a kind now ranks ahead of a straight flush or royal flush. Is this mathematically correct? "

On a frequency basis (of course) it is not mathematically correct -- you know that.... But it is a definition for the game at hand and you must play by the rules of the house (or play someplace else). Years ago, for instance in San Diego CA poker rooms the joker(bug) was used in draw poker games. The house rules was that the joker could be used as a wild card for a straight, a flush, a straight flush and also used as an ACE, But the house rule was that if you had five aces it would lose to a straight flush. It was a rule definition in San Diego. The important thing is to know the house rules when you play in unusual places.

My advice is to try to get a feeling for the effect of four wild cards in a game. It usually takes at least four of a kind and big cards to take down a nice pot. If you don't have at least one wild card -- than get out early. In stud, straight flushes are common with four wild cards. I would not play in a flop game with wild cards.

How many wild cards are there?

If there is only 1 wild card there are only 13 ways to make 5 of a kind.

And there are 40 natural straight flushes.

And there are 11 straight flushes in each suit of the form (n,n+1, n+2,
n+3, Joker), 10 of the form (n,n+1, n+2 Joker, n+4), 10 of the form (n,n+1, n+3 Joker, n+4)
and 10 of the form (n,n+2, n+3 Joker, n+4). So that's 41 in each suit or 164 total.

So in this case a straight flush should lose to 5 of kind as it is a less common hand.