Rare Hand of Quads plus Badugi

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.

So that is 52 * 36 * 22 * 10. But you have to account for different orders by dividing by 4 factorial.
(52 * 36 * 22 * 10) / 4! = 17,160.
Then for each of those hands, you have a choice of 4 cards to make quads and fill in the remainder of the hand of seven cards.
4 * 17,160 = 68,460.
Other way – start with quads and make a badugi.
13 ways to make quads. You need to pick up another 3 cards.
 For any of those, you can pick any suit next, but not match rank – 4 * 12. Then you have 3 suits to pick from with 11 ranks available, 3 * 11. Then you have choice of two suits, and 10 ranks, 20. Again, because order does not matter, you need to divide by 3 factorial.
(13 * 48 * 33 * 20)/3! = 68,640.
Checks out!
Now, the number of 7 card hands that make a straight flush can be found in lots of places, such as the Wizard of Odds. That number is 41,584.
As crazy as Bill’s hand was, a straight flush is rarer.

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).


Author: Sleepy

Eyes droop and he peers from meer slits
Long gone the usual insight and wit
then a raise unexpected
the pot is collected
How long will he keep up this shit?

Leave a Reply