In Thursday night’s game while playing Boardugi (7-card stud variant split high-low with a badugi to qualify for low – added degeneracy of a royalty if you can make a badugi with cards showing on the board), we had the pleasure of being scooped by Dollar Bill’s amazing hand of four-of-a-kind for high, with a badugi for the low.
The question came up – is that rarer than a straight flush?
So let’s do the math.
We will do it two ways, to see we get the same answer.
First off – begin with a badugi and then combine the quads.
To make a badugi, you can start with any of 52 cards, then you need one of the remaining not the same suit (3) * not the same rank (12), or 36 possible. Then you need one of the other remaining suits (2) * the remaining ranks (11), or 22. Then you have 10 cards that help you make a badugi.
But not by much.
Note: another remarkable thing was that Bill’s badugi was pretty good. Better than a 10-low. If we just look at the number of ways to make a ten low with quads, we get (10 * 36 * 24 * 14) / 3! = 20,160.
Now that is rarer than a straight flush, but not quite as rare as a royal (4,324 combinations).