| | Ivars Peterson's MathTrek |
September 9, 1996
The standard game and its many variants involve a curious mixture of luck and skill. Given a deck of 52 cards, there are 2,598,960 ways to select a subset of five cards. So, the probability of getting any one hand is 1 in 2,598,960.
One of the first things a novice player learns is the relative value of different sets of five cards. At the top of the heap is the straight flush, which consists of any sequence of five cards of the same suit. There are 40 ways of getting such a hand, so the probability of being dealt a straight flush is 40/2,598,960, or .000015.
The next most valuable type of hand is four of a kind, and so on. The table below lists the number of possible ways that different types of hands can arise and their probability of occurrence.
| Rankings of Poker Hands and Their Frequencies of Occurrence: | ||
| Hand | No. of Ways | Probability |
|---|---|---|
| Straight flush | 40 | .000015 |
| Four of a kind | 624 | .000240 |
| Full house | 3,744 | .001441 |
| Flush | 5,108 | .001965 |
| Straight | 10,200 | .003925 |
| Three of a kind | 54,912 | .021129 |
| Two pair | 123,552 | .047539 |
| One pair | 1,098,240 | .422569 |
The rules of poker specify that a straight flush beats four of a kind, which tops a full house, which bests a flush, and so on through a straight, three of kind, two pair, and one pair. Whatever your hand, you can still bet and bluff your way into winning the pot, but the ranking (and value) of the hands truly reflects the probabilities of obtaining various combinations by random selections of five cards from a deck.
Many people, however, play a livelier version of poker. They salt the deck with wild cards -- deuces, jokers, one-eyed jacks, or whatever. The presence of wild cards brings a new element into the game, allowing a player to use such a card to stand for any card of the player's choosing. It increases the chances of drawing more valuable hands.
It also potentially alters the ranking of different hands. One can even draw a five of a kind, which typically goes to the top of the rankings.
Just how much wild cards alter the game is recounted in an article in the current issue of Chance, written by mathematician John Emert and statistician Dale Umbach of Ball State University in Muncie, Ind. They analyze wild-card poker and conclude, "When wild cards are allowed, there is no ranking of the hands that can be formed for which more valuable hands occur less frequently."
That's a striking result. Wild cards increase the number of ways in which each type of hand can occur. The amount of that increase depends on which cards are designated as wild. For example, with deuces wild, four of a kind occurs more than twice as often as a full house. So, modifying the rules to rank a full house higher than four of a kind might produce a more consistent result.
A player, however, often has a choice in how to declare a hand, and that choice will invariably produce the strongest possible combination according to the declared rules. Thus, if a full house ranks higher than four of a kind, and a player has a wild card allowing him or her to choose either a full house or four of a kind, the full house will inevitably come up more often than four of a kind!
"There is no possible ranking of hands in wild-card poker that is based solely on frequency of occurrence," Emert and Umbach convincingly demonstrate. The authors examined several wild-card options and found that the standard ranking proves to have fewer inconsistencies than other possible ranking schemes.
They then went on to see if there exists a better way of ranking the hands, proposing a scheme that takes into account the fact that certain hands can be labeled in several ways. For example, any wild-card hand declared as a full house can also be considered as two pair, three of a kind, or even one pair or four of a kind.
The authors define a quantity called the inclusion frequency, which gives the number of five-card hands that can be declared as such for each type of hand. Rankings based on this number give hands with smaller inclusion frequencies a higher position in the list. In standard poker, this method leads to the traditional rankings. Wild-card variants show a slightly different order. Interestingly, one result of this new ranking criterion is that the greater the number of wild cards, the more valuable a flush becomes.
"We believe that the use of the 'inclusion' ranking of the hands presents a more consistent game than deferring to ordinary ranking," Emert and Umbach declare.
Of course, this analysis doesn't really take into account the complexity of what actually happens in a poker game. You're not likely to be computing probabilities as you play. It may be much more advantageous for you to put on your best poker face and bluff as much as you think you can get away with.
In an analysis of simple games that involve bluffing, John Beasley, in The Mathematics of Games, wryly counsels: "Do not think that a reading of this chapter has equipped you to take the pants off your local poker school. Three assumptions have been made: that you can bluff without giving any indication, that nobody is cheating, and that the winner actually gets paid. You will not necessarily be well advised to make these assumptions in practice."
There are some aspects of poker that are beyond the reach of mathematics.
Emert, John, and Dale Umbach. 1996. Inconsistencies of "wild-card" poker. Chance 9(No. 3):17-22.
Packel, Edward W. 1981. The Mathematics of Games and Gambling. Washington, D.C.:Mathematical Association of America.
John Emert can be reached at emert@math.bsu.edu.
For a glimpse of the world of serious poker enthusiasts, you can start at the web site: http://www.panix.com/~ssf/poker.html.
Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.