How does choice affect the odds of forming ranked five-card poker hands?

Most poker games are resolved on the basis of a five-card-hand hierarchy. Common sense suggests that rankings within the hierarchy track the odds against forming the various hands. That is, royals should be the toughest to get, followed by lower-order straight flushes, quads, full houses, and so forth. But, what are those odds, how do they change with rank, and to what degree are they affected by the choices afforded to players during the course of alternate games?

The most familiar poker formats fall into three categories:

• Hands are fully formed by five randomly drawn cards, such that players are purely at the mercy of fate. Examples are the classic five-card stud, as well as Caribbean Stud and Let-It-Ride – reasonably popular house-banked casino games.
• Hands are initially formed by five randomly drawn cards, but players can elect to replace one or more (up to a maximum set by the rules of the game in question); the substitute cards are also drawn at random. Examples are five-card draw and most versions of video poker.
• Hands are formed by choosing five of seven randomly drawn cards. Examples are seven-card stud and Texas hold ‘em.

The accompanying table shows the odds against finishing with a hand at various levels of the five-card hierarchy in each of the indicated classes of games. The data for five- and seven-card stud apply to all games in the respective categories. The figures for five-card draw depend on the maximum permissible number of substitutions and also on the criteria solid citizens employ to choose what to hold and what to dump; the odds presented in this column of the table are those for the optimum strategy in a particular video poker payout configuration.

Odds against making ranked five-card hands in alternate versions of poker

Hand               five-card stud     five-card draw     seven-card stud

Royal flush        649,739.0-to-1      40,386.7-to-1       30,939.6-to-1
Straight flush      72,192.3-to-1       9,147.3-to-1        3,589.5-to-1
Four of a kind       4,164.0-to-1         422.3-to-1          594.0-to-1
Full house             693.2-to-1          85.9-to-1           37.5-to-1
Flush                  507.8-to-1          89.8-to-1           32.1-to-1
Straight               253.8-to-1          88.1-to-1           20.6-to-1
Three of a kind         46.3-to-1          12.4-to-1           19.7-to-1
Two pair                20.0-to-1           6.7-to-1            3.3-to-1
One pair                 1.4-to-1                               1.3-to-1
High pair                5.2-to-1           3.7-to-1
Low pair                 2.8-to-1		
High card                1.0-to-1                               4.6-to-1
High card or low pair                       0.8-to-1

With one exception, the cited odds to be overcome get longer as hands move up the standard hierarchy. In the version of video poker used to illustrate the five-card draw game; a full-house pays more but is easier to make than a flush or straight. This reversal applies for optimum strategy with most, although not all, payout schedules in nothing-wild jacks-or-better video poker.

The changes in odds between hands at successive levels of the hierarchy are somewhat counterintuitive. The progressions are highly nonuniform. In five-card stud games, royals are 9.0 times less frequent than non-royal straight flushes; this result can be expected because nine non-royals are possible in each suit (from ace-low through nine-low) while only one royal can be formed (10-low). In seven-card stud games, the shift in odds is similar, royals being 8.6 times more rare than non-royals. In five-card draw games, however, the chances of the two classes of straight flushes are much closer to one another – the multiplier being a smaller 4.4.

The large jump in all three configurations when moving down the list from straight flushes to quads often surprises players. As do the relatively minor differences among straights, flushes, and full houses – along with the big gap between these hands and four-of-a-kind.

Comparisons across games are revealing as well. They show that the element of choice has major consequences. They also indicate that picking-and-choosing in draw games has a lesser influence on reducing the odds against high-ranking hands than does selecting the best five-out-of-seven in stud variations.

Other probability considerations are also important in poker. Most notably, critical decisions often hinge on the likelihood a hand will improve by discarding and drawing, or as additional cards are revealed. At video poker and most house-banked games, the strategies you can learn by rote already account for such factors. Facing other players in “live” poker, well, that’s when you need to heed this rhetorical rhyme from the sapient scribe, Sumner A Ingmark:

What more enhances your finances,
Than knowledge of inherent chances?