Fourflush/fourstraight Frequency

[Al Mirpuri]

Anyone, I need to know how often a fourflush/fourstraight gets dealt within five cards.

I used to have this information in a booklet but no more.

Any help appreciated.

[MarkGritter]

Four-flushes are easy. There are 13 cards in each suit. There are 13C4 ways to pick four of them = 715 four-flushes per suit. The fifth card in the hand should not be of the same suit, so there are 39 different choices. That gives 4 * 715 * 39 = 111,540 four-flushes in 2,598,960 five-card hands, or about 4.3%.

If you want to restrict things to just four-flushes without pairs, then the 5th card can be one of the 27 remaining after eliminating suited and pairing cards to the four-flush, so 77,220 possible four-flushes without pairs, about 3%.

Four-straights are harder, because we presumably want to count just those that aren't four-flushes as well.

There are 11 possible four-straights (A234 through JQKA). Only 9 of them are open-ended, of course. Do you want to count the gutshots or just 2345 through TJQK? I'll work through the latter.

There are four choices for each card in the four-straight. There are 8 cards that make up a straight. So, ignoring flushes, there are:

9 * 4^4 * (52-4-8) = 92,160 open-enders (including open-enders with pairs)

How many ways can we have a flush or four-flush? Well, we could have a staight flush draw in just one way per straight/suit.

9 straights * 4 suits * (52-4-8) 5th cards not completing a straight = 1,140 straight flush draws (possibly including made flushes)

Or, we could have three suited cards as part of the flush, and the fourth card completing a four-flush but not the straight, for example:

5c6c7h8cXc, X != 4 or 9, nor 7 because we already counted that case above

9 straights * 4 suits * 4 ways to choose three of the suit * (13-4) suited cards = 1,296 flush draws which are not straight-flush draws.

So, the total number of OESDs that aren't four-flushes, including ones with pairs, is
89,724, about 3.45%.

I'm too lazy to work out how to eliminate pairs.

[Al Mirpuri]

For a non-mathematician, as I am (think it cool but am hopeless) this is fantastic! Much appreciated.

[SteelWheel]

fwiw, there is a five-card stud variant, which I have seen referred to variously, as "Soko" or "Canadian Stud" or "Scandinavian Stud". Dealing and betting rounds are same as normal five-card stud, i.e., Round 1=one down, one up, Round 2=second up card, Round 3=fourth up card, Round 4=fifth and final up card (although not sure if this game calls for a low card force, or a high card force).

Anyway, the hand ranks are the same as any other poker game, with the addition of four-flushes and four-straights. These are ranked in between two pair and one pair. I.e., two pair >four-flush > four-straight > one pair. Seems like it would be an interesting game, as there is now more room for bluffing opportunities on the end.

[Al Mirpuri]

Thanks for the post SteelWheel. I am familiar with Canadian Stud (though never played it). It is being called Soko these days because it is popular in Finland (?) and that is what they call it.

New York stud is five stud with only the fourflush counting.