Hack 47. Go Crazy with Wild Cards

Wild cards are added to a poker game to ratchet up the fun. Statistically, though, they make things all discombobulated.

Hundreds of years ago, poker players settled on a rank order of hands and decided what would beat what. Pleasantly, for the field of statistics, the order they settled on is a perfect match with the probability that a player will be dealt each hand. Presumably, the developers of poker rules either did the calculations or referenced their own experience as to how frequently they saw each kind of hand in actual play. It is also possible that they took a deck of cards, paper and pencil, and a free afternoon, dealt themselves many thousands of random poker hands, and collected the data. Whatever the method, the rank order of poker hands is a perfect match with the relative scarcity of being dealt those particular combinations of cards.

Rank ordering, though, does not take into account the meaningful distance between one type of hand and the type of hand ranked immediately below it. A straight flush, for example, is 16 times less likely to occur than the hand ranked immediately below it, which is four of a kind, while a flush is only half as likely as a straight, the hand ranked immediately below a flush.

Before we talk about the problem with playing with wild cards (cards, often jokers, that can take on any value the holder wishes), let's review the ranking of poker hands. Table 4-17 shows the probability that a given hand will occur in any random five cards, as well as each hand's relative rarity when compared to the hand ranked just below it in the table.

Table Poker hands, probabilities, and comparisons

Hand	Probability	Relative rarity
Straight flush	.000015	16 times less likely
Four of a kind	.00024	5.8 times less likely
Full house	.0014	1.4 times less likely
Flush	.0019	2.1 times less likely
Straight	.0039	4.4 times less likely
Three of a kind	.021	2.3 times less likely
Two pair	.048	8.8 times less likely
One pair	.42	1.2 times less likely
Nothing	.50	-----

To gamblers, there are several observations of note from Table 4-17. First, with five cards, half the time players have nothing. Almost half the time, a player has a pair. A player will have something better than a pair only 8 percent of the time.

Second, some hands treated as if they are wildly different in rarity are almost equally likely to occur. Notice that a flush and a full house occur with about the same frequency.

Finally, after three of a kind, the likelihood of a better hand occurring drops quickly. In fact, there are two giant drops in probability: having either nothing or a pair occurs most of the time (92 percent), then two pair or three of a kind occurs another 7 percent of the time, and something better than three of a kind is seen less than 1 percent of the time.

The Problem with Wild Cards

This is all very interesting, but what does it have to do with the use of wild cards? Well, adding wild cards to the deck screws up all of these time-tested probabilities. Assuming that the holder of a wild card wishes to make the best hand possible, and also assuming that one wild card, a joker, has been added to the deck, Table 4-18 shows the new probabilities, compared to the traditional ones.

Table Probability of poker hands with one wild card in the deck

Hand	Probability with wild card	Classic probability	Change in probability with wild card
Five of a kind	.0000045	-----	-----
Straight flush	.000064	.000015	+327 percent
Four of a kind	.0011	.00024	+358 percent
Full house	.0023	.0014	+64 percent
Flush	.0027	.0019	+42 percent
Straight	.0072	.0039	+85 percent
Three of a kind	.048	.021	+129 percent
Two pair	.043	.048	-10 percent
One pair	.44	.42	+5 percent
Nothing	.45	.50	-10 percent

The problem with wild cards is apparent as we look at the new probabilities, especially when we look at three of a kind and two pair. Three of a kind is now more common than two pair!

The rank order that traditionally determines which hand beats what is no longer consistent with actual probabilities. Additionally, the chances of getting two pair actually drop when a wild card is added. Other probabilities change, of course, with all the other playable hands becoming more likely. Some super hands, while remaining rare, increase their frequency quite dramatically: hands better than three of a kind are about twice as common as they were before.

Knowing these new probabilities gives smart poker players an edge. In fact, contrary to the stereotype that experienced and professional poker players avoid games with wild cards because they are childish or for amateurs, some informed players seek out these games because they believe they have the advantage over your more naïve types. (You know, those naïve types, like people who don't read Hacks books?)

Why It Works

As you can see in Table 4-18, using wild cards lessens the chance of getting two pair. But why would this be? Surely adding a wild card means that sometimes I can turn a one-pair hand into a two-pair hand. This is true, but why would I? Imagine a player has one pair in her hand, and she gets a wild card as her fifth card. Yes, she could match that wild card up with a singleton and call it a pair, declaring a hand with two pairs. On the other hand, it would be smarter for her to match it up with the pair she already has and declare three of a kind. Given the option between two pair and three of a kind, everyone would choose the stronger hand.

The Other Problem with Wild Cards

The existence of wild cards creates a paradox that drives game theorists crazy. The paradox works like this:

1. The ranking of hands and their relative value in a poker game should be based on the frequency of their occurrence. The less frequently occurring hand should be valued more than more commonly occurring hands.

2. In the case of choosing whether to use a wild card to turn a hand into two pair or three of a kind, players will usually choose to create three of a kind. This changes the frequency in practice such that two pair becomes less common than three of a kind.

3. Because rankings should be based on probabilities, the rules of poker should be changed when wild cards are in play to make two pair more valuable than three of a kind.

4. With revised rankings, three of a kind would be worth less than two pair, so now smart players would use their wild card to make two pair instead of three of kind, so two pair would quickly become more common than three of a kind.

5. The ranking rules would then have to be changed again to match the actual frequencies resulting from the previous rule change, and a never-ending cycle would begin.

Table 4-18 avoids this paradox by assuming that players want to make their best hand based on traditional rankings. Clever of me, huh? Want to play cards?