Possible Resolution of Wild Card Poker Paradox

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 hands 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.

Inclusion Frequencies with corresponding Hand Ranking
Two Jokers			One Joker			No Wilds
Ranking	Frequency		Ranking	Frequency		Ranking	Frequency
5 of K.	78		5 of K.	13
Str. Fl.	624		Str. Fl.	204		Str. Fl.	40
4 of K.	9,438		4 of K.	3,133		4 of K.	624
Flush	12,012		Flush	8,008		F. H.	3,744
F. H.	18,174		F. H.	9,061		Flush	5,148
Straight	35,328		Straight	20,736		Straight	10,240
3 of K.	256,494		3 of K.	146,965		3 of K.	59,280
2 pair	325,094		2 pair	215,605		2 pair	127,920
1 Pair	1,844,366		1 Pair	1,551,797		1 Pair	1,281,072

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. “

Conclusion

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.

References and Further Reading

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.

Appendix 1

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 …

Appendix 2

Calculations for hands with one joker wild:

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.