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# How to Compute the Probability of a Flush in Stud Poker

In this lesson, we explain how to compute the probability of being dealt an ordinary flush or a straight flush in stud poker. (For a brief description of stud poker, click here.)

## What is a Flush?

In stud poker, there are two types of hands that can be classified as a flush.

• Straight flush. Five cards of the same suit in sequence, such as 3, 4, 5, 6, 7.
• Ordinary flush. Five cards of the same suit, not in sequence, such as 2, 6, 8, 9, Q.

In this lesson, we will compute probabilities for both types of flush.

## How to Compute Poker Probabilities

In a previous lesson, we explained how to compute probability for any type of poker hand. For convenience, here is a brief review:

• Count the number of possible five-card hands that can be dealt from a standard deck of 52 cards
• Count the number of ways that a particular type of poker hand can occur
• The probability of being dealt any particular type of hand is equal to the number of ways it can occur divided by the total number of possible five-card hands.

So, how do we count the number of ways that different types of poker hands can occur? We recognize that every poker hand consists of five cards, and the order in which cards are arranged does not matter. When you talk about all the possible ways to count a set of objects without regard to order, you are talking about counting combinations. Luckily, we have a formula to do that:

Counting combinations. The number of combinations of n objects taken r at a time is

nCr = n(n - 1)(n - 2) . . . (n - r + 1)/r! = n! / r!(n - r)!

In summary, we use the combination formula to count (a) the number of possible five-card hands and (b) the number of ways a particular type of hand can be dealt. To find probability, we divide the latter by the former.

## Probability of a Straight Flush

Let's execute the analytical plan described above to find the probability of a straight flush.

• First, we count the number of five-card hands that can be dealt from a standard deck of 52 cards. This is a combination problem. The number of combinations is n! / r!(n - r)!. We have 52 cards in the deck so n = 52. And we want to arrange them in unordered groups of 5, so r = 5. Thus, the number of combinations is:

52C5 = 52! / 5!(52 - 5)! = 52! / 5!47! = 2,598,960

Hence, there are 2,598,960 distinct poker hands.
• Next, we count the number of ways that five cards can be dealt to produce a straight flush. A straight flush consists of five cards in sequence, each card in the same suit. It requires two independent choices to produce a straight flush:
• Choose the rank of the lowest card in the hand. For a straight, the lowest card can be an ace, 2, 3, 4, 5, 6, 7, 8, 9, or 10. So, we choose one rank from a set of 10 ranks. The number of ways to do this is 10C1.
• Choose one suit for the hand. There are four suits, from which we choose one. The number of ways to do this is 4C1.

The number of ways to produce a straight flush (Numsf) is equal to the product of the number of ways to make each independent choice. Therefore,

Numsf = 10C1 * 4C1 = 10 * 4 = 40

Conclusion: There are 40 different poker hands that fall in the category of straight flush.
• Finally, we compute the probability. There are 2,598,960 unique poker hands. Of those, 40 are straight flushes. Therefore, the probability of being dealt a straight flush (Psf) is:

Psf = 40 / 2,598,960 = 0.00001539077169

The probability of being dealt a straight flush is 0.00001539077169. On average, a straight flush is dealt one time in every 64,974 deals.

## Probability of an Ordinary Flush

The Venn diagram below shows the relationship between a straight flush and an ordinary flush. Everything within the rectangle is a flush, in the sense that it is a poker hand with five cards in the same suit. The blue circle is an ordinary flush; the red circle, a straight flush. You can tell that a straight flush and an ordinary flush are mutually exclusive events, because the circles do not intersect or overlap. And because the events are mutually exclusive,

Pf = Psf + Pof

where Pf is the probability of any type of flush, Psf is the probability of a straight flush, and Pof is the probability of an ordinary flush. To compute the probability of an ordinary flush, we rearrange terms, as shown below:

Pof = Pf - Psf

From the analysis in the previous section, we know that Psf = 0.00001539. Therefore, to compute the probability of an ordinary flush (Pof), we need to find Pf. Here is how to find the probability of an ordinary flush:

• First, count the number of five-card hands that can be dealt from a standard deck of 52 cards. We did this in the previous section, and found that there are 2,598,960 distinct poker hands.
• Next, count the number of ways that five cards from a 52-card deck can be arranged to produce a flush. It requires two independent choices to produce a flush:
• Choose the rank of each card in the hand. A playing card can have a rank of 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king, or ace. That's 13 distinct ranks. So, we choose five ranks from a set of 13 ranks. The number of ways to do this is 13C5.
• Choose one suit for the hand. There are four suits, from which we choose one. The number of ways to do this is 4C1.

The number of ways to produce a flush (Numf) is equal to the product of the number of ways to make each independent choice. Therefore,

Numf = 13C5 * 4C1

Numf = 1,287 * 4 = 5,148

Conclusion: There are 5,148 different poker hands that can be classified as a flush - either a straight flush or an ordinary flush.
• Finally, compute the probability of being dealt a flush. There are 2,598,960 unique poker hands. Of those, 5,148 are some form of flush. Therefore, the probability of being dealt a flush (Pf) is:

Pf = 5,148 / 2,598,960 = 0.001980792317

Now, we can find the probability of being dealt an ordinary flush. It is:

Pof = Pf - Psf

Pof = 0.001980792317 - 0.00001539077169

Pof = 0.001965401545

where Pf is the probability of any type of flush, Psf is the probability of a straight flush, and Pof is the probability of an ordinary flush.

Bottom line: In stud poker, the probability of an ordinary flush is 0.0019654. On average, it occurs once every 509 deals.