There have been many conversations about which games should pay out for straight flushes. Setting aside wildcard and variable deal games (e.g. pass em, push em, buy happiness) it actually is reasonably easy to lay out the relative chances of the straight flush by determining the number of five card hands that can be made in each game. Each possible five card hand has an equal chance of being a straight flush, so the number of possible five card hands directly corresponds to the probability of a straight flush in the game.

The absolute probabilities for straight flushes actually vary slightly from the numbers below (due to the possibility of 6 or 7 card straight flushes occurring) but for the purposes of comparing the games the method is valid.

Note that the chances of a five card hand being a straight flush is one in 64,974.

Overview

Stud poker is a straight forward deal of 7 cards to each player of which five must be played. This gives 21 possible hands that can be played (math is below). Comparatively some of the common games have the following number of possible hands:

Game	Possible Hands	Approximate Chance of Straight Flush
5-5 Buy 1	6	10,829:1
Stud	21	3,094:1
5-4 Buy 2	21	3,094:1
Cross	41	1,585:1
Butthole	48	1,354:1
The Judge	51	1,274:1
Omaha	60	1,083:1
42nd Street	72	902:1
Colinoscopy	81	802:1
Two Touch	101	643:1

Despite what might have occurred in the past few months Butthole is actually lower than Omaha and only about twice as likely as Stud. Colinoscopy and Two Touch (as far as I recall) haven’t been challenged as valid games for paying the straight flush bonus.

Math on Stud

The 21 combinations is most easily understood by looking at the hand as two cards having to be discarded rather than 5 retained. There are 7 choices for the first card to be discarded and six for the second, and 7×6 is 42. However since choosing card A then B has the same effect as B then A the 42 actually is double counting the number of options, leading to 21 possible pairs of discards, leaving 21 cards as the total number of hands that could be played from 7 cards.

Common Math

(And if you don’t trust my math, check out the entry in Wikipedia)

There are some common calculations that crop up so rather than repeating them in multiple places I’ll state them here:

2 cards from 5 has 10 combinations

3 cards from 5 has 10 combinations

2 cards from 4 has 6 combinations

Math on Other Games

5-5 Buy 1

This one is fairly straightforward – you can keep the five dealt cards (1) or discard one of five and receive a replacement (5).

1+5 = 6

5-4 Buy 2

Effectively a seven card hand, so same as Stud

Cross

The fun begins. Three options – use one, two or three from the cross.

1 from the cross (5 possibilities) means all four cards from the hand must be played (1) – 1×5 = 5

2 from the cross (6 possibilities) means three of four cards from the hand (4) – 6×4 = 24

3 from the cross (2 possibilities) means two from four from the hand (6) – 2 x 6 = 12

5+24+12 = 41

Butthole

There are eight choices for the common cards, and two from four from the hand (6) must be used.

6×8 = 48

Judge Getz More

Two choices, play the entire hand (1) or three from the hand (3 of 5 = 10) and a pile (5).

1+ 10×5 = 51

Omaha

Must be three of five common cards (10) and two of four from the hand (6).

10×6 = 60

42nd Street

Must be two of four from the hand (6), two of four from the top line (6) and one of 2 from the bottom line (2). 6x6x2 = 72

Colinoscopy

Three options:

Five from the hand (1), zero from the common (1) = 1

Four from the hand (5), one from the common (4) = 5×4 = 20

Three from the hand (10), two from the common (6) = 10×6 = 60

1+20+60 = 81

Two Touch