Reconciling wild cards and poker

CHANCE 17

Reconciling Wild Cards and PokerKristen A. Lampe

Human behavior often defies logic. Consider, for example, the success of state lotteries, despite sta-tistics published demonstrating that a person has a

greater chance of being struck by lightning than winning a major prize. Another illustration of illogical human behavior is the popularity of playing poker with wild cards, both at home and online.

Drawn in by the excitement generated by higher-ranked hands occurring more frequently, people ignore the warnings in John Emert and Dale Ulbach’s 1996 CHANCE article and Stephen Gadbois’ 1996 Mathematics Magazine article of math-ematical inconsistencies created by using wild cards in poker. Without wild cards, poker hands are ranked in a mathemati-cally satisfying way: the scarcer the hand, the higher the rank. The crux of the problem with wild cards in poker is that once added to the game, wild cards change the likelihood of each hand. The mathematical property of higher-ranked hands being less probable is suddenly lost.

The articles mentioned above explore why this mathemati-cal property is lost, but do not suggest any solutions. Here, we propose a rule for ranking hands with wild cards so higher-ranked hands remain less probable. This rule is easy to implement and maintains the excitement provided by the higher hands occurring more often when wild cards are in use.

Poker BackgroundWe are focusing on standard fi ve-card hands, with various additions or declarations of wild cards. Here, “rank” refers to the type of hand and “denomination” refers to the value of an individual card. In poker, once a ranking of hands is established as part of the rules, a higher-ranked hand beats a lower-ranked one. Without wild cards, the standard ranking of poker hands is as follows:

Straight FlushFour of a KindFull HouseFlushStraightThree of a KindTwo PairOne PairHigh Card

A straight fl ush is defi ned as cards in sequence sharing the same suit. Ace can be high or low in the straight, but straights

18 VOL. 24, NO. 4, 2011

Hand Frequency

Straight Flush 40

Four of a Kind 624

Full House 3,744

Flush 5,108

Straight 10,200

Three of a Kind 54,912

Two Pair 123,552

One Pair 1,098,240

Table 1—Ranked Standard Poker Hand Frequencies with 52-Card Deck

do not “wrap around,” so A2345 of hearts is a straight fl ush, but QKA23 of spades is not.

Throughout, we will use a, b, c, d, and e to denote dif-ferent denominations of cards, so four of a kind is a hand having form aaaab, where b is commonly referred to as the kicker. A full house has form aaabb, while a straight is fi ve cards in a sequence having at least one card of a different suit than the others. Three of a kind looks like aaabc, two pair has the form aabbc, and one pair has the form aabcd. The high card hand has fi ve different denominations without forming a straight or a fl ush. Table 1 lists the frequencies for these hands, using a standard 52-card deck and no wild cards. Notice how the frequency increases as the rank of the hand decreases.

Playing poker with wild cards nearly dates back to the origins of modern poker, itself. In the early 1800s, a 20-card

deck was most common. The game involved four players, each dealt fi ve cards. Straight and fl ush hands were not yet recognized. By the mid 1800s, the game evolved into using the standard 52-card deck to accommodate more players and provide enough cards for a “draw,” a wrinkle on poker intro-duced about this time.

Popularity exploded during the Civil War, and the hands expanded to include the straight and flush. Initially, the straight was ranked between three of a kind and two pair, but gradually found its way to its proper mathematical spot in the rankings. For about 30 years, four of a kind remained the highest-ranking hand, beating any straight fl ush hands that had yet to be distinguished from straight or fl ush hands. It took some time, but eventually the hands became ranked in the manner to which we are now accustomed.

Interestingly, poker players were unswayed by mathematical arguments regarding the scarcity of a straight fl ush, preferring to keep the rankings as they were (a predictor, perhaps, of today’s wild card conundrum). A now-standard “moral argument” that straight fl ush must be ranked above four of a kind appeals to the “ungentlemanly” aspect of betting on sure-to-win hands of four aces plus a kicker or four Kings plus an ace. According to

David Parlett’s The Oxford Guide to Card Games, use of wild cards is referenced as early as 1875 in a variation

of poker using the blank card accompanying a deck of cards as the wild card.

The Wild Card ConundrumAs mentioned earlier, Emert, Ulbach, and Gadbois show that adding wild cards creates mathematical diffi culties. In fact, these diffi culties were noted even earlier by masters of the game, such as Parlett and John Scarne, without any resolutions suggested. The standard technique of playing

with wild cards is fi rst to provide a ranking of hands. Each poker hand is

CHANCE 19

then declared to be the highest possible hand within the ranking. Thus, with the standard ranking, a hand Waaab, where W denotes a wild card, is declared to be four of a kind. A new rank of fi ve of a kind is added, found only using wild cards. Table 2 provides frequencies using a 54-card deck with two Jokers added as wild cards, as well as frequencies with a 52-card deck declaring deuces wild as calculated by Emert and Ulbach.

We can see the diffi culties in adding two wild Jokers: Three of a kind has become much more likely than two pair (a lower-ranked hand). Four of a kind and full house have the same likelihood. In short, the frequency does not increase as the ranks decrease. A less serious problem is that fi ve aces is now an unbeatable hand.

In previous papers, the suggestion was made to alter the ranking of hands. The problem that arises is that once the ranking is provided and the hand declared to be the high-est possible within that ranking, certain hands are declared differently. For example, if we reverse the ranking of three of a kind and two pair, declaring two pair the better hand (a logical move), the 164,736 hands of the form Waabc that had been three of a kind switch allegiance to two pair. This tips the balance and now two pair, the higher-ranked hand, is more likely than three of a kind. Emert, Ulbach, and Gadbois conclude there is no way to rank the hands in order for the frequencies to increase as the ranks decrease once wild cards are added to the deck.

The problem seems fairly clear. Adding wild cards or declaring cards within the deck to be wild cards, and then using them to form hands of highest possible rank, creates mathematical inconsistencies. Even after changing the ranking of hands (and mathematicians love to change the rules to see what happens), the inconsistencies do not go away. How-ever, these issues do not seem to mitigate the excitement of having the higher-ranked hands occur more frequently, and wild card usage has certainly not disappeared from poker

games. The most cursory of web searches verifi es this. In fact, most video poker sites have wild card games such as Deuces Wild or One-Eyed Jacks whose payout system has similar fl aws as face-to-face wild card poker. Namely, the payout increases as the rank of hand increases, but the way the hands are ranked is not in decreasing order of probability. (The reader is invited to explore whether the expected value of playing such a game outweighs the cost of the game.) The observation that people persist in playing poker with wild cards leads us to desire reconciliation between use of wild cards and the proper progression of probabilities for the hands.

The Wild Card RuleSince we mathematicians are not above changing the rules of a game, we fi rst propose the following procedure: Rank the hands in the standard manner, with fi ve of a kind being the highest-ranked hand. For hands with no wild cards in them, assign them to the highest possible ranking as usual. For hands involving wild cards, fi nd the highest two possible ranks and assign the hand the lower of the two. For example, the hand Waabc has three of a kind and two pair as its highest two possible ranks and will thus be assigned a two pair rank. In all originality, we will call this the Wild Card Rule. Table 3 pro-vides a full accounting using the Wild Card Rule with Jokers or deuces wild, but it is worthwhile to examine a few hands to understand the process of assigning and counting.

Counting Hands with Two Jokers Wild and the Wild Card RuleWhen adding two Jokers to the deck, counting frequencies using the Wild Card Rule is fairly straightforward for fi ve of a kind, straight fl ush, four of a kind, and full house. For example, the only way to get fi ve of a kind would be a hand of the form Waaaa. This hand can be declared fi ve of a kind or four of a

Two Jokers Wild Four Deuces Wild

Hand Frequency Hand Frequency

Five of a Kind 78 Five of a Kind 672

Straight Flush 624 Straight Flush 2,552

Four of a Kind 9,360 Four of a Kind 31,552

Full House 9,360 Full House 12,672

Flush 11,388 Flush 14,472

Straight 34,704 Straight 62,232

Three of a Kind 232,968 Three of a Kind 355,080

Two Pair 123,552 Two Pair 95,040

One Pair 1,437,936 One Pair 1,225,008

Table 2—Frequencies with Jokers or Deuces Wild Using Standard Ranking

20 VOL. 24, NO. 4, 2011

Two JokersWild Four Deuces Wild

Hand Count Hand Count


Straight Flush 40 Straight Flush 80

Four of a Kind 702 Four of a Kind 1,808

Full House 12,480 Full House 34,192

Flush 5,692 Flush 5,000

Straight 10,200 Straight 8,160



One Pair 1,124,296 One Pair 801,968

Table 3—Using the Wild Card Rule with Jokers or Deuces Wild

kind or lower, depending on the denomination assigned to W. The Wild Card Rule then declares this hand to be four of a kind, and thus there are zero fi ve of a kind hands.

The fi rst interesting hand to consider, then, is the fl ush. There are 5,108 natural fl ush hands. For the one wild card hand of the form Wabcd to be declared a fl ush, it must have straight fl ush and fl ush as the two highest possible declarations. Thus, we need to count those hands with higher designation straight fl ush. The components to count are the number of ways to assign the lowest nonwild denomination (a); the number of ways to pick W; the number of ways to place b, c, and d among the four remaining places; and the number of ways to assign a suit. Notice the counting differs if W is used as the lowest card in the straight fl ush, so it must be considered separately, giving a total of 328 fl ush hands with one wild card. (We invite the reader to examine the full counting argument in the supplementary online material at http://chance.amstat.org/category/supplemental.) Hands of the form WWabc whose higher designation is a straight fl ush are counted in a similar manner for a total of 256. Thus, there are 5,692 total fl ush hands under the Wild Card Rule.

Let’s explore one more scenario from the two Jokers wild category. To count the three of a kind hands, fi rst note there are 54,912 natural hands. Under the Wild Card Rule, hands of the form Waabb are declared three of a kind, and there are 5,616 of them. With two wild cards, those hands of the form WWabc whose higher designation is a fl ush or a straight are now declared three of a kind. There are 888 hands whose higher designation is a fl ush (but not a straight fl ush). Counting those hands whose higher designa-tion is a straight is more involved (see http://chance.amstat.org/category/supplemental). There are 3,840 hands of this type, giving a total of 65,256 three of a kind hands.

It is worth noting that, with any declaration of wild cards, all wild card hands with one pair as the highest rank are

declared high-card hands. Thus, the number of high-card hands remains higher than that of any other rank.

Notice from Table 3 that the Wild Card Rule doesn’t quite work right away. The frequency of full house hands is now greater than that of fl ush or straight hands. This happens because the wild card hands that have higher ranking four of a kind are declared full house hands with the Wild Card Rule, and there are many more of these than wild card hands that move to fl ush or straight hands.

To fi x this, let’s redefi ne the ranking of hands, moving full house below fl ush or straight. Notice how this change in ranking does not affect the assignments of hands. Those hands assigned to be full house under the Wild Card Rule with the original ranking, whose higher ranking is four of a kind, cannot be fl ush or straight hands in the new ranking because there must be a natural pair in the hand. Therefore, there is no change in how any of the hands are assigned in Table 4. So, we now have a ranking of hands and a rule to declare hands that results in a mathematically consistent game for two Jokers wild.

Counting Hands with Deuces Wild and the Wild Card RuleThe case of declaring deuces wild is slightly more complicated and deserves some attention. Notice in Table 3 that a similar problem occurs when fi rst using the Wild Card Rule with the standard ranking of hands. Again, full house occurs with higher frequency than fl ush or straight. After re-ranking the hands placing full house lower than straight, however, some of the hands will be assigned differently. But, the problem isn’t too severe; only one type of hand has this issue. Hands of the form WWWab have higher designation of either straight fl ush or four of a kind, depending on a and b. In the fi rst case, the Wild Card Rule still assigns all such hands as four of a kind

CHANCE 21

hands. When the highest rank is four of a kind, however, we no longer automatically assign all hands to full house with the revised ranking of hands. Some may be flush or straight hands.

Thus, of the 3,568 hands whose higher rank is a four of a kind, we need to determine how many are declared fl ush, straight, or full house hands under the Wild Card Rule. A hand becomes a fl ush hand if a and b are the same suit but not close enough for a straight fl ush, so we have 400 hands of this type. This count chooses the three Ws, assigns denomination to a and

b, and picks a suit, subtracting the number of straight fl ush hands (see http://chance.amstat.org/category/supplemental). To determine how many are declared a straight, note that a and b must be close enough for a straight, but not have the same suit. After methodical counting, one concludes there are 1,968 hands declared straights under the Wild Card Rule. Finally, 3,568 - 400 - 1,968 = 1,200 hands of this type are declared to be full house.

The second column of Table 5 gives the frequencies of hands using the Wild Card Rule with deuces wild. Notice that, once again, we have a mathematically consistent game.

Four Deuces Wild, Original Rankings

Four Deuces Wild, Revised Rankings




Four of a Kind 1,808 Four of a Kind 1,808

Full House 34,192 Flush 5,400

Flush 5,000 Straight 10,128

Straight 8,160 Full House 31,824



One Pair 801,968 One Pair 801,968

Table 5—The Wild Card Rule, Deuces Wild, Before and After Re-Ranking

Two Jokers Wild, Original Rankings Two Jokers Wild, Revised Rankings




Four of a Kind 702 Four of a Kind 702






One Pair 1,124,296 One Pair 1,124,296

Table 4—The Wild Card Rule, Jokers Wild, Before and After Re-Ranking

22 VOL. 24, NO. 4, 2011

Two Jacks Wild, Original Rankings Two Jacks Wild, Revised Rankings




Four of a Kind 624 Four of a Kind 624






One Pair 923,072 One Pair 923,072

Table 6—The Wild Card Rule, Two Wild Jacks, Before and After Re-Ranking

Counting Hands with One-Eyed Jacks Wild and the Wild Card RuleThe fi nal case we consider is declaring one-eyed Jacks (and we assume there are two such Jacks in any deck) to be wild. As you might suspect, the most diffi cult hands to count are straight fl ush and straight hands, but even in hands such as four of a kind, we must consider the cases involving Jacks separately, as there are only two nonwild Jacks remaining. For example, to count the hands whose higher rank is a four of a kind, the natural hands have the form aaaab. This means a cannot be a Jack, so there are

such hands. With one wild card, Waaab,

again a cannot be a Jack and we get

hands, choosing the wild card, a’s denomination, a’s suits, and b, respectively. With two wild cards, however, WWaab allows a to be a Jack and that case must be considered separately.

There are then hands of

this type, for a total of 8,328 four of a kind hands. Implement-ing the Wild Card Rule will lead us to assign the Waaab and WWaab hands as full house hands.

Counting the hands whose higher denomination is straight fl ush will illuminate the method for counting straights as well. The natural hand abcde has two cases. If a is ace through 6, we have 6 * 1 * 4 = 24 ways to choose the placement of a, the placement of b through e, and the suit. If a is 7 to 10, we have 4 * 1 * 2 = 8 ways to make these choices, for a total of 32 natural straight fl ush hands. When we consider hands with one wild card, Wabcd, the four cases to consider are:

a is ace to 6

a is 7 to 10 and b is a Jack

a is 7 to 10 and there is no nonwild Jack

a is a nonwild Jack

In that order, there are

hands of this type. For two wild cards, hands of the form WWabc have the same four cases as above, with the additional case of a being a Queen. There are

hands of this form. Thus, there are a total of 534 straight fl ush hands without the Wild Card Rule. Using the Wild Card Rule, all wild card hands above are assigned the rank of fl ush. Table 6 gives the frequencies of hands with two Jacks used as wild cards. Again, we obtain a mathematically consistent game.

Next GameWe began by examining a problem arising when wild cards are used in poker. Using the traditional method of declaring every hand the best possible hand with the standard poker rankings, the mathematical progression of higher-ranked hands occurring with less frequency was lost. As determined in earlier papers, even changing the ranking of hands does not remove this mathematical inconsistency. However, if we changed the rules of declaring the hands with wild cards to the second-best ranking hand, we could modify the standard hand rankings to produce a game in which frequencies increase as rankings decrease.

We have explored the outcomes of using the Wild Card Rule in three of the most common assignments of wild cards.

CHANCE 23

In each case, the result is a mathematically consistent game, necessitating only a slight modifi cation in the ranking of hands from the standard game of poker. In the case of deuces wild, the Wild Card Rule leads to interesting recalculations—as would be the case any time there are three or more wild cards. The reason the number of three wild cards is minimal is that, in this case, hands can be declared four of a kind, full house, fl ush, and/or straight. With two wild cards, there is no possibility of a hand occurring that could be both a four of a kind and a fl ush or straight, because a natural pair is necessary.

One interesting consequence of the Wild Card Rule is that no hand is ever declared fi ve of a kind. This suits us just fi ne (no pun intended) for two reasons. First, it was a new hand to the traditional list of poker hands. Second, when using fi ve of a kind, there is a guaranteed unbeatable hand of fi ve aces. As mentioned earlier, certain experts in the game consider betting on such a hand rather rude. Thus, we use fi ve of a kind as a type of placeholder here, and we are quite happy about that.

Another interesting consequence of the Wild Card Rule is an added level of wildness to a game involving draw cards. For example, if a player held W2689 of clubs, the Wild Card Rule declares this to be a lowly pair of 9s. In a game of fi ve card draw, the player is tempted to turn in the wild card in hopes of another club. Will this play give the player the best chance of a better hand?

A colleague remarked that were he to get the hand 10JQKA of hearts in a one-eyed Jacks wild game with a one-eyed Jack of hearts, he would be crushed not to be able to declare such a hand a straight fl ush. That is, he would like the option of treating that one-eyed Jack of hearts as a plain old Jack of

Hearts instead of a wild card. Since there aren’t many occasions when a player might want a wild card to be nonwild, we end with an interesting question: If the wild cards were once originally part of a standard deck, must we use them as wild cards every time, or will the Wild Card Rule remain mathe-matically consistent if the wild cards are not mandatory wild cards, but can be used as nonwild cards if it is to the player’s advantage? We leave this for the reader to explore.

Further Reading

Emert, J., and D. Umbach. 1996. Inconsistencies of ‘wild-card’ poker. CHANCE 9:17–22.

Gadbois, S. 1996. Poker with wild cards—a paradox? Math-ematics Magazine 69:283–285.

McManus, J. 2009. Cowboys full: The Story of poker, Farrar, Straus, and Giroux. Macmillan: New York, NY.

Parlett, D. 1990. The Oxford guide to card games. Oxford Univer-sity Press: Oxford, UK.

Scarne, J. 1979. Scarne’s guide to modern poker. Simon and Schus-ter: New York, NY.

24 VOL. 24, NO. 4, 2011

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Reconciling wild cards and poker