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This is not the most exciting paradox you will ever come across at curiouser.co.uk. However it is one of the less well known paradoxes, and it should be of interest to anyone who plays poker or has a mathematically curious mind for the paradoxical. In poker, hands are ranked according to their frequency – the lower the probability of receiving a hand, the higher the hand is ranked. When wild cards are introduced, it becomes impossible to rank the hands according to their frequency; there will always be at least one hand that is ranked below a hand of higher frequency. For a normal deck of 52 cards the frequencies/probabilities are easily calculated and result in the following hierarchy of hands:
See appendix 1 for calculations “Wild cards” are sometimes introduced to a deck in order to liven up a game. These cards (which are often jokers but can be any of the standard 52 cards) can then be used to count as any card, thus increasing the probability of making a high-ranking hand.However, by introducing wild cards a paradox is created: Let us assume that we introduce one joker as a wild card (WC) and retain the standard hierarchy of hands as tabled above. If one were to be dealt 2D 2C 6S 8H WC, one would have a choice as to whether to call the WC an 8 (making 2 pair) or a 2 (making 3 of a kind). Obviously, one would choose whichever hand had the highest rank, which in this case would be three of a kind, but by doing this, one actually makes frequency of a two pair hand less frequent than a Three of a Kind hand, which ought to rank two of a kind above three of a kind in the hierarchy. However, if one changes the ranking in this way, so that two of a kind ranks above three of a kind, three of a kind immediately become less frequent again and should therefore be ranked above Two of a Kind. No matter whatever hierarchy of hands is introduced, the probabilities/frequencies are always incompatible. The following table shows the standard hierarchy of 5 card poker hands with their probabilities of occurrence. |
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The table below shows what happens to the frequencies when a joker is introduced to the deck as a wild card.
| Frequency for Two Rankings with One Joker Wild | ||||
| Ranking | Frequency | Ranking | Frequency | |
| 5 of K. | 13 | 5 of K. | 13 | |
| Str. Fl. | 184 | Str. Fl. | 184 | |
| 4 of K. | 3,120 | 4 of K. | 3,120 | |
| F. H. | 6,552 | F. H. | 6,552 | |
| Flush | 7,804 | Flush | 7,804 | |
| Straight | 20,532 | Straight | 20,532 | |
| 3 of K. | 137,280 | 2 Pair | 205,920 | |
| 2 Pair | 123,552 | 3 of K. | 54,912 | |
| 1 Pair | 1,268,088 | 1 Pair | 1,268,088 |
The number of possible hands has now increased to 53C5 = 2,869,685. However, the declaration of a hand which contains a wild card (and therefore the hand’s frequency) will now depend on the ranking. For example: if one holds 2C 2H 6D JS WC, one could either declare the wild card to be a J or a 2 depending on whether three of a kind or two pair had the higher rank.
See appendix 2 for calculations and further explanation.
Although not shown on here, similar anomalies arise using 2 Jokers, “Deuces Wild” and other wild card variations. An interesting observation is that when using two jokers wild, if Four of a Kind is ranked above a Full House (as it is naturally), the frequency of both hands is exactly the same (9360 ways; probability = 0.0030). It is also interesting to note that the more wild cards used, the lower the frequency of flushes becomes, as four of a kind and full houses become more common.
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One possible solution to this paradox was proposed by John Emert and Dale Umbach 1. Recognising that it was necessary to rank hands for wild card poker, they proposed a new system that they called “inclusion frequency”: “We propose a ranking that, rather than partitioning the ranked hands into disjoint [sic] categories, acknowledges that certain hands can be labeled in several ways. For example, any hand that could be labeled a full house could also be considered as two pair, three of a kind, or even one pair. A hand such as (Waaab) could be declared to be any of these types as well as a four of a kind… We define ‘inclusion frequency’ for each type of hand to be the number of five card hands that may be declared as such. The inclusion frequency ranking is determined by ranking the hands so that those type of hands that have smaller inclusion frequencies are more valuable.” Using this system, a four of a kind hand will always be ranked higher than a three of a kind because a four of a kind could also be declared as a three of a kind plus two other cards. Equally a three of a kind would also be counted as a possible two of a kind for the purpose of the inclusion frequency ranking, insuring that three of a kind will always be ranked above two of a kind. The following table show how the hierarchy of hand changes using this system of ranking. You will note that there are no inconsistencies between the rank of the hand the inclusion frequencies, although using both one and two jokers a flush now ranks ABOVE a full house. (Although not shown in this table, when using two deuces wild, using the inclusion frequency method a flush ranks above both a full house and four of a kind.) It is also interesting to note that that with no wild cards, using the inclusion frequency ranking the hierarchy is exactly the same as with standard frequency ranking.
Emert and Umbach note that “one interesting facet of this [inclusion frequency] ordering principle is that the greater the number of wild cards, the more valuable flush becomes. The other types of hands seem to keep their ‘natural’ ranking among themselves. “
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It would seem that the “inclusion frequency” method of ranking hands for wild card poker offers a neat mathematical solution to this paradox. However, poker is a game of simple rules and complex judgements. To change the ranking of hands for wild card poker may resolve the paradox and offer a certain consistency but it would not improve the game.
Understanding the effect that the introduction of wild cards has on the probabilities in poker will undoubtedly be advantageous to any player. In Doyle Brunson’s Super System he notes that many professional poker players like to play with wild cards because their knowledge of the altered probabilities gives them an extra advantage over their opponents.
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1J. Emert, D. Umbach, "Inconsistencies of 'Wild Card' Poker",
Chance Magazine, 9(1996), 17-22.
S. Gadbois, "Poker with wild cards--- a paradox?", Math. Mag. 69(1996), pp. 283-285
Epstein, R. A. (1977), The Theory of Gambling and Statistical Logic, New York: Academic Press.
Findler, N. V. (1978), “Computer Poker” Scientific American, 239, 144-151
Litwiller, B. H., and Duncan, D. R. (1977) “Poker Probabilities: A New Setting,” Mathematics Teacher, 70, 766-771.
Calculations for hands with no wild cards:
The number of ways that 5 cards can be dealt from a deck of 52 cards is 52 C5 = (52!)/(5!47!) = 2,598,960
Of these hands, how many are 4 of a kind? Four of a kind requires 4 cards of the same denomination plus any one of the remaining 48 cards. Because the choice of the extra card is independent from the choice denomination of the other four, the calculation is 13 * 48 = 624. (Alternatively: 13 C1 4 C4 * 48 C1 = 13*1*48 = 624)
The probability of being dealt Four of a Kind from a deck of 52 cards is therefore 624/2,598,960 = 0.000240
How many ways are there to get a full house? A full house requires 3 cards of one denomination and 2 cards of another. The matching triple can be any of 13 the denominations and the pair can be any of the remaining 12 denominations. There are 4 C3 = 4 ways to select the suits of the matching triple and 4 C2 = 6 ways to select the matching pair. As these selections are independent, the calculation for the number of ways to select a full house from a pack of 52 cards is 13 * 4 * 12 * 6 = 3,744.
The probability of being dealt a full house from a pack of 52 cards is therefore 3,744/2,598,960 = 0.001441
The probabilities of being dealt the other hands can be calculated using similar methods:
Three of a kind:
13 C1 * 4 C3 * 12 C2 * 4 C1 * 4 C1 = 13 * 4 * 66 * 4 * 4 = 54912
Two Pair:
13 C2 * 4 C2 * 4 C2
* 44 C1 = 123,552
etc …
5-of-a-kind:
13C1 * 4C4 * 1 = 13
probability = 13/2869685 = .00000453
Three-of-a-kind when ranked ABOVE Two Pair:
(This can either be dealt naturally, as above, or with one pair plus the wild joker)
54912 + 13 C1 * 4 C2 * 1 * 12 C2 * 4 C1 * 4 C1 = 54912 + 13 * 6 * 66 * 4 * 4 = 137280
Probability = 137,280/2,869,685 = .04784
Two Pair when ranked BELOW Three of a Kind:
(Two pair can only be dealt naturally, because if one were dealt One Pair plus a joker, it would be declared as Three of a Kind not Two Pair.)
As calculated above, there are 123,552 get be dealt Two Pair.
Probability = 123,552/2,869,685 = .043054
Note that this probability is LESS than the probability of being dealt Three of a Kind.
When Two Pair is ranked ABOVE Three of a Kind:
(Using the same calculations as above)
123,552 + 13 C1 * 4 C2 * 1 * 12 C2 * 4 C1 * 4 C1 = 205,920
Probability = 205,920/2,869,685 = 0.071757
Three-of-a-kind when ranked BELOW Two Pair:
(This hand can only be dealt naturally as one pair plus a wild card would be declared as Two Pair.)
Using the calculation above, number of ways = 5,4912
Probability = 5,4912/2,869,685 = 0.0191352
Note that this probability is now LOWER than Two Pair, despite (and because of) being ranked lower in the hierarchy of hands.
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