Poker originated in the Louisiana territory around the year 1800. Ever since, this addictive card game has preoccupied generations of gamblers. It has also attracted the attention of mathematicians and statisticians.
The standard game and its many variants involve a curious mixture of luck and skill. Given a deck of 52 cards, you have 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.
A novice poker player quickly learns the relative value of various 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.
The table below lists the number of possible ways that desirable hands can arise and their probability of occurrence.
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 whatnot. The presence of wild cards brings a new element into the game, allowing such a card to stand for any card of the player's choosing. It increases the chances of drawing more valuable hands.
Using wild cards also potentially alters the ranking of various sets of cards. You can even obtain five of a kind, which typically goes to the top of the rankings.
A while ago, mathematician John Emert and statistician Dale Umbach of Ball State University took a close look at wild-card poker. Wild cards can alter the game considerably, they wrote in a 1996 article in Chance describing their findings. "When wild cards are allowed, there is no ranking of the hands that can be formed for which more valuable hands occur less frequently," the authors argued.
In other words, when you play with wild cards, you can't rely on a ranking of hands in the order of the probability that they occur as you can when there are no wild cards. Magician and card expert John Scarne made a similar observation in his book Scarne on Cards, first published in 1949.
In a 1996 article in Mathematics Magazine, Steve Gadbois also concluded that wild cards mess up logical, probability-based poker play, producing all sorts of anomalies or paradoxes in the ranking of different hands. "The more one looks, the worse it gets," he remarked.
Wild cards increase the number of ways in which each type of hand can occur. 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 of how to declare a hand and that means assembling the strongest possible combination allowed by the given 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 demonstrated. The researchers also examined alternative ranking schemes. They found that whatever the wild-card option, the standard ranking proves to have fewer inconsistencies than the alternatives.
Emert and Umbach then went on to see if there exists a better way of ranking the hands. They proposed 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 declared.
Of course, these analyses don'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 discussion 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."
Some aspects of poker are beyond the reach of mathematics.
Beasley, John D. 1989. The Mathematics of Games. Oxford, England: Oxford University Press.
Emert, John, and Dale Umbach. 1996. Inconsistencies of "wild-card" poker. Chance 9(No. 3):17-22.
Gadbois, Steve. 1996. Poker with wild cards—A paradox? Mathematics Magazine 69(October):283-285.
Packel, Edward W. 1981. The Mathematics of Games and Gambling. Washington, D.C.: Mathematical Association of America.
Scarne, John. 1991. Scarne on Cards. New York: New American Library.
For information on poker odds, see "Poker Odds for Dummies" (Cardschat.com).