Last week I wrote about the odds and probabilities of every five card poker hand. If you missed the article you can read it here.
This week, I’m going to look at how these odds change if we add a wild card into the mix.
A wild card, sometimes called a joker, can be used to represent a card of any value and suit. This can cause some interesting effects (as we'll see later). It also means that five of kind is possible.
Because the wild card can be used to represent any card, we’re going to make the assumption that the player uses the wild card to maximum advantage. What this means is that, if the player is dealt a wild card, s(he) uses it in a position that results in the highest possible ranking hand.
A consequence of this is that you will never see the hand two pairs including a wild card. If you have a wild card, you would be better off making three of a kind from the natural pair. (And if a hand was two natural pairs and a wild card, it would automatically be a full house). Similarly, the lowest possible hand including a wild card is a pair. (It’s impossible to have a high card hand with a wild card, as this wild card will automatically pair with the highest natural card). 

Let's refresh ourselves on the basic odds. To the left is a table showing the frequency of all possible five card poker hands. The way these figures were derived are explained in last weeks article. There are 52 cards in a standard deck and so _{52}C_{5} possible sets of cards, resulting in a total of 2,598.960 possible combinations. The addition of a wild card adds an additional card to the set _{53}C_{5} = 2,869,685. 
Let's repeat the exercise from last week of going through all hand rankings and finding out number distinct sets of cards that make each of these groups when there is one wild card in the deck. 
There are only 13 sets of cards in the entire deck that can result in five of a kind. The wild card must be present, and the other four cards need to be quads. There is only one way to make five of a kind for each of the thirteen values in the deck.
13 five of a kind hands.
There are three different ways to make a straight flush when there is, potentially, a wild card.
The first is the natural way (no wild card involved). Once the top card of a straight flush is defined, the rest of the cards follow on automatically. A straight flush can be have an Ace as the top card, or a King, or Queen … down to the five. There are ten possible straight flush hands per suit, and four suits, so 40 possible natural straight flushes.
When a wild card is added, things are a little more complicated. A wild card can be used to fill any possible position of a straight flush, except the lowest card of the straight (because if there were four cards in a row the player would select the wild card to be at the top end of the straight; remember, we're assuming the player makes the highest possible hand). The exception to this is when the straight is Ace high. In this case, the wild card can be any of the cards, even the lowest one.
Let's deal with the latter special case first. For an ace high straight flush (a royal flush), the wild card can be any of the five cards: A,K,Q,J,T and there are four suits, so there are 20 ways this can happen.
For an nonroyal straight flush there are nine possible high ending cards, and the wild card can be any one of the four higher cards, and there are four suits, so with a middle wild card there are: 9 × 4 × 4 = 144.
40 + 20 + 144 = 204 Straight flush hands.
There are two ways to make four of a kind. The first is a natural way, and the second is with a triple and a wild card. Because we're assuming the player always attempts to make the highest ranking hand, they will use the wild card to make a quad with the triple, and not to pair up with the singleton to make a full house.
In the natural case, there are thirteen different cards that can be used for the quads: A,K,Q,J,T … 3,2, this leaves 48 other cards for the singleton (not 49, because we can't use the wild card as this would be five of a kind). So the number of possible hands is 13 × 48 = 624.
If one wild card is involved, there are thirteen values for what the triple can be, and this can be made _{4}C_{3} different ways (all the ways four cards can be arranged in groups of three). Then there is the wild card, and the remaining singleton must be one of the 48 remaining cards that is not the same as the triple. So the number is: 13 × 4 × 48 = 2,496
624 + 2,496 = 3,120 Four of a kind hands.
There are two ways to make a full house. The natural way, and with two pairs and a wild card.
For a natural full house there is a triplet and a pair. The triplet can be one of any of the 13 card values, and three out of the four possible cards chosen. The pair will come from one of the twelve other cards and there are the combinations of the ways that these pairs can be made from four cards.
The calculation for the natural full house is: _{13}C_{1} × _{4}C_{3} × _{12}C_{1} × _{4}C_{2} = 13 × 4 × 12 × 6 = 3,744.
For the two pairs, there are 13 possible values for the the first pair, and 12 possible values for the second pair, but, as the player will want to maximize the hand, the wild card will be used only to triple up the higher pair, so the number of combinations is half this (13 × 12) ÷ 2 = 78. For each of these combinations of numbers there are _{4}C_{2} ways the suits can be arranged. So the number of full houses with a wild card is: 78 × 6 × 6 = 2,808.
3,744 + 2,808 = 6,552 Full House hands.
As before there are two ways to make a flush. The natural way, and with a wild card.
We have to remember that, when counting flushes, we must subtract the special case of flushes that are also straight flushes (to avoid double counting). As 'cards speak' a hand that is also a straight flush will have been counted earlier. We've already determined that there are 204 straight flushes.
For natural flushes there are _{13}C_{5} ways to pick five cards of the same suit, and four different suits: 1,287 × 4 = 5,148 natural flushes.
With a wild card, there are _{13}C_{4} ways to pick the other cards, and still four suits: 715 × 4 = 2,860 flushes with one wild card.
5,148 + 2,860  204 = 7,804 Flush hands.
In just the same was as with the straight flush, there are three cases to consider with the regular straight. The first is the natural way. The second is with a wild card being in any location but the last value in the straight, and finally a wild card in an acehigh straight (where the wild card can be in any of the positions).
Once we have the totals for all of these we must remember to subtract away the number of straight flush hands (204) as we do not want to double count them.
For a natural straight, there are 10 possible values for the high card, and each of the five cards can be any one of the four suits: 10 × _{4}C_{1}^{5} = 10 × 1,024 = 10,240.
When there is a wild card and the top card is not an ace, this can happen nine times. Each nonwild card can be one of four suits, and the wild card can be any one of the lower four positions: 9 × _{4}C_{1}^{4} × 4 = 9 × 256 × 4 = 9,216.
The last case is when there is wild card and the straight is ace high (also called by the friendly name "broadway"). In this case there are four other cards that can be any suit, and the wild card can be in any of the five positions: _{4}C_{1}^{4} × 5 = 256 × 5 = 1,280.
10,240 + 9,216 + 1,280  204 = 20,532 Straight hands.
There are two ways for get three of a kind. The natural way and using a wild card to triple up a natural pair.
The natural way requires three cards to be one of the thirteen values, and these three cards can be selected from four suits. The final two cards are selected to be different from what is left: _{13}C_{1} × _{4}C_{3} × _{12}C_{2} × (_{4}C_{1})^{2} = 13 × 4 × 66 × 16 = 54,912.
For the wild card case, there are 13 possible values for the pair (in various combinations of suits), then the wild card, then the two cards needed from the remaining 12 and each of these can be from any suit: _{13}C_{1} × _{4}C_{2} × _{12}C_{2} × (_{4}C_{1})^{2} = 13 × 6 × 66 × 16 = 82,368.
54,912 + 82,368 = 137,280 Three of a kind hands.
The only way for two pairs to be made is a natural two pair. (If a wild card showed up with one pair, a better hand could be made by converting these to three of a kind, so the wild card would never be used to form the second pair).
There are thirteen possible values that each of the natrual pairs could have, and from these we need to select two _{13}C_{2}. Each of these pairs can be made from two differenent suits _{4}C_{2}. The singleton can have a value of any of the other unused 11 cards _{11}C_{1}, and be any one of the suits _{4}C_{1}
Two Pair: _{13}C_{2} × (_{4}C_{2})^{2} × _{11}C_{1} × _{4}C_{1} = 78 × 36 × 11 × 4 = 123,552.
123,552 Two pair hands.
This hand is not possible if there is a wild card for the obvious reason that if a wild card existed then, at a minimum, it would pair with the at least one of the other cards. There are, therefore, the same frequency of occurences as naturally occuring high card hands (see previous article if you want the derivation).
1,302,540 High card hands.
There is one natural way to make a pair, and one way in which the wild card pairs up with the highest singleton in the hand.
We're doing this calculation last and out of order because this is the most complicated rank to calculate. We can apply the principle of complemenet and logical subtraction. As we know there are a total of _{53}C_{5} possible hands, we can subtract off all the other calculated ranked hands and get the number of one pair hands. _{53}C_{5} = 2,869,685
One pair = 2,869,685  13  204  3,120  6,552  7,804  20,532  137,280  123,552  1,302,540 = 1,268,088
1,268,088 Pair hands.
Here is a table of the results with one wild card:
Hand  Frequency 

Five of a kind  13 
Straight Flush  204 
Four of a kind  3,120 
Full House  6,552 
Flush  7,804 
Straight  20,532 
Three of a kind  137,280 
Two pair  123,552 
One pair  1,268,088 
High card  1,302,540 
TOTAL  2,869,685 
If you look carefully you will see that frequency of two pair is less than three of a kind (making it rarer). This creates a dilema. What we should do is rank the hands in relation to how rare they are; this would require us to switch the ordering of two pair to make it rank higher than three of a kind. But now the situation is crooked the other way around! If a player is dealt a wild card and a pair, he would forgo the three of a kind to make two pair. This means that the only way three of a kind would be possible is from a natural way, but now that makes it less likely again (as we'd remove 82,368 ways from three of a kind, to two pair, and return three of a kind to just 54,912 ways). 
There is an irreconilable dilemma for the ordering of two pairs and three of a kind when there is one wild card.
(This dilemma is solved in the "Pai Gow" variant of poker where the wild card can only be used in the completion of straights and flushes. Outside of this it is given the value of an ace. This reverts the ordering of two pair and three of a kind to their natural order).
This situation gets worse if more wild cards are added. Here are various other ways that poker is often "spiced up" with the addition of other wild cards:
Sometimes games are played with two wild cards (two jokers). This deck contains 54 cards.
Sometimes "oneeyed" Jacks are considered wild (there two of these Jacks in a deck, so there are still 52 cards, but now two are wild). Here the rankings are effected in otherways reducing the relative probabilities of natural straights, but increasing probabilities of others.
Sometimes games are played "Two's wild". (Here there are still 52 cards, but now every two is wild for a total of four wild cards).
Below are tables of the frequencies of these odds:
Natural  One Joker  Two Jokers  OneEyed Jacks  Two's Wild  

Five of a kind  0  13  78  72  672 
Straight flush  40  204  624  534  2,552 
Four of a kind  624  3,120  9,360  8,328  31,552 
Full House  3,744  6,552  9,360  8,112  12,672 
Flush  5,108  7,804  11,388  9,476  14,472 
Straight  10,200  20,532  34,704  28,938  62,232 
Three of a kind  54,912  137,280  232,968  199,758  355,080 
Two pairs  123,552  123,552  123,552  102,960  95,040 
Pair  1,098,240  1,268,088  1,437,936  1,188,652  1,225,008 
High Card  1,302,540  1,302,540  1,302,540  1,052,130  799,680 
Wild Cards  0  1  2  2  4 
Total Cards  52  53  54  52  52 
Total  2,598,960  2,869,685  3,162,510  2,598,960  2,598,960 
Take aways:
With two wild cards, the disparity between three of a kind and two pairs increases.
Also, with two wild cards, a pair becomes more common than a high card.
With two wild cards, four of a kind, and a full house, occur with the same frequency.
With Oneeyed Jacks wild, four of a kind becomes more common than a full house.
With all the two's wild, four of a kind becomes quite common, and the number of high card hands reduces even though no wild cards are used in these hands (because it becomes impossible to have a sevenhigh hand with no natural twos).
Similarly, with oneeyed Jacks wild, the number of natural Jackhigh hands is reduced.
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