The Odds of Poker

Permutations and Combinations

The first goal in determining poker odds is this: We need to know how many possible hands can be dealt in a normal 52 card deck of 4 suits and 13 ranks. It is easy to think that there are 52 possible cards for the first card, 51 for the second, 50 for the third and so on, and then we derive the following formula : 52*51*50*49*48=311,875,200 possible poker hands.

This is a good start, but not accurate. This is a standard permutation,which looks like this:

Remember that little exclamation point is the factorial symbol, which is defined as:

The Permutation is fine, except that in poker, the order of the hands delt is not important, so we use a different formula, similar to the Permutation formula, called the Combination formula, which looks like this:

So there are 2,598,960 possible poker hands dealt. We will assign this value to the number N.

Ranks of Poker Hands

The Rank of Poker Hands are as follows:

High Card (Ace is the highest rank, 2 is the Lowest, 7 is the lowest “high card”)
One Pair (two cards of the same rank and three cards of different rank)
Two Pair (four cards in pairs and the fifth card of a third rank)
Three of a Kind (three cards of equal rank)
Strait (Five cards of progressive ranks)
Flush (all five cards are in the same suit)
Full House
Four of a Kind
Straight Flush
Royal Flush

We’ll start at the easiest figures to calculate, the higher ranked hands.

Royal Flush

Lets Start at the easiest, which is the Royal FlushA Royal Flush is the A-K-Q-J-10 in the same suit. There are only four suits, so there are only four possible Royal Flushes on a deal of N possible hands, so:

Straight Flush

Now lets look at the Straight Flush, which is 5 cards of the same suit in consecutive order. A Royal Flush is the Highest Ranking Straight Flush, so we will have to ignore those four hands.The possible runs are K-Q-J-10-9 to 5-4-3-2-A, so in each suit there are 9 runs (not including the Royal) and four suits, so 36 Straight Flushes are possible. The odds are:

Four of a Kind

Next Hand we will look at is the Four of A Kind. A Four of A Kind is just that, four cards of the same rank with a fifth card in your hand. There are 13 Ranks, so we have 13 possible hands, plus a fifth card. There are 48 possible cards to fill that fifth space. so there are 13 x 48 = 624 possible hands for a Four of a kind.

Full House

The Full House is a combination of three cards of the same rank and a pair in another rank.To find how many possible three card combinations there are, we need to know how many ways we can have, say three out of four aces, or twos, or whatever. We need ₄C₃, which is 4, and 13 ranks, so we have 13*4 or 52 ways of getting a three of a kind. For the remaining two cards, we need to know how many combinations of 2 cards in any rank, so ₄C₂, which is 6, spread out over the 12 remaining ranks, so there are 72 possible pairs. This gives us:

So far we have accounted for 4,408 possible hands, the chance of getting one of these four hands is 0.001696063 or less that one tenth of a percent, so don’t fret if you rarely get one of these hands.

Flush

The next highest ranking hand is the Flush, which is when all five cards are in the same suit. First, we need to figure out how many possible ways we can get five cards in one suit, so the ₁₃C₅, which is 1,287. This must be multiplied by the four suits, so 1,287 x 4 = 5,148. So far, so good, but this number includes the straight flushes and the royal flush. Subtract 36 for the total number of Straight Flushes and 4 for the total number of Royal Flushes, and we get:

Straight

Next we have the Straight, where five cards of consecutive rank but not in the same suit are in hand. I’ll be honest, this one I can’t figure out. The given formula is:

This is obviously wrong, since this means there are less Straights than Four of a Kinds, even though Straits are about 16 times more common...so, mulitplying 0.003924647 by 2,598,960 = 10,200 straights. The given formula must be wrong.
Taking 10,200, adding the 40 back in that are subtracted, we have 10240, diving by 10 (since we can make sense of that number) we are left with 1024. This is a recognizable number. Since we’ve accounted for the number 10 as being the 10 ranks that can start a run, all we have to do is figure out, starting at one rank, how many possible Straights we have. Lets start at 5, there are 4 Fives, 4 Fours, 4 Threes, 4 Twos, and 4 Aces, or 4^5 which is, (fortunately) 1024.
So the original formula is a typo, it should appear:

Three of a Kind

Next, we have the Three of a Kind which is three cards of the same rank and two cards of two different ranks. The first part is easy, recall from the Full House equation 13*(₄C₃)=52, this determines the probabile ways of getting three cards of the same rank, and now the two sluff cards. After dealing three cards, there are 48 possibilities for the fourth card (52 minus the four in the main suit) and 44 possibilities for the fifth card (52 minus 8 for the 2 suits we’ve already had). Now we need to look at the possible combinations for the last two cards, we have 48 x 44, and there are two possible ways for these cards to appear, so the final equation is:

Two Pair

The last two equations are for Two Pair and One Pair, and so far we have covered 54,912 + 10200 + 5108 + 3744 + 624 + 36 + 4 = 74,628 out of 2,598,960, which is 0.028714563% or just under 3% of the possible poker hands.
Two Pair occur when you have two cards of one ranks, two cards of a second rank, and one card of a third rank. The first pair is familiar: 13(₄C₂) and the second pair is 12(₄C₂), or 78 and 72, or 5,616. x 44 = 247104

So there is another 5 percent of poker hands, roughly.

A Pair

The last hand is a Pair, two out of five cards are worth something.Finding the probabilities of the pair is easy : 13(₄C₂), we’ve seen this a couple of times. The last three cards are found like this:

So the final equation looks like:

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