### Probability

#### Probability Basics

#### Probability Problems

#### Poker Probability

- Probability in stud poker
- Probability of straight
- Probability of flush
- Cards of equal rank
- Probability of no pair

#### Random Variables

#### Discrete Distributions

#### Continuous Distributions

### Probability: Table of Contents

#### Probability Basics

#### Probability Problems

#### Poker Probability

- Probability in stud poker
- Probability of straight
- Probability of flush
- Cards of equal rank
- Probability of no pair

#### Random Variables

#### Discrete Distributions

#### Continuous Distributions

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

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

## What is a Straight?

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

**Straight flush**. Five cards of the same suit in sequence, such as 3♥, 4♥, 5♥, 6♥, 7♥.**Ordinary straight**. Five cards in sequence, with at least two cards of different suits. Ace can be high or low, but not both. Thus, A♠, 2♥, 3♦, 4♣, 5♥ and 10♠, J♥, Q♦, K♣, A♥ are valid straights; but Q♠, K♥, A♦, 2♣, 3♥ is not.

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

## 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

_{n}C_{r} = 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:

Hence, there are 2,598,960 distinct poker hands._{52}C_{5}= 52! / 5!(52 - 5)! = 52! / 5!47! = 2,598,960 - 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
_{10}C_{1}. - Choose one suit for the hand. There are four suits, from which we choose one.
The number of ways to do this is
_{4}C_{1}.

The number of ways to produce a straight flush (Num

_{sf}) is equal to the product of the number of ways to make each independent choice. Therefore,Num

Conclusion: There are 40 different poker hands that fall in the category of straight flush._{sf}=_{10}C_{1}*_{4}C_{1}= 10 * 4 = 40 - 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
- 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 (P
_{sf}) is:P

_{sf}= 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 Straight

The Venn diagram below shows the relationship between a straight flush and an ordinary straight. Everything within the rectangle is a straight, in the sense that it is a poker hand with five cards in sequence. The blue circle is an ordinary straight; the red circle, a straight flush.

Notice that the circles do not intersect or overlap. From that, you can infer that a straight flush and ordinary straight are mutually exclusive events. Therefore,

P_{s} = P_{sf} + P_{os}

where P_{s} is the probability of any type of straight, P_{sf} is the probability of a straight flush, and P_{os} is the
probability of an ordinary straight. To compute the probability of an ordinary straight, we rearrange terms, as shown below:

P_{os} = P_{s} - P_{sf}

From the analysis in the previous section, we know that the probability of a straight flush (P_{sf}) is 0.00001539077169. Therefore, to compute the probability of
an ordinary straight (P_{os}), we need to find P_{s}. Here is how to find P_{s}:

- 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 in sequence.
It requires six independent choices to produce a straight:
- 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
_{10}C_{1}. - Choose one suit for the first card in the hand. There are four suits, from which we choose one.
The number of ways to do this is
_{4}C_{1}. - Choose one suit for the second card in the hand. There are four suits, from which we choose one.
The number of ways to do this is
_{4}C_{1}. - Choose one suit for the third card in the hand. There are four suits, from which we choose one.
The number of ways to do this is
_{4}C_{1}. - Choose one suit for the fourth card in the hand. There are four suits, from which we choose one.
The number of ways to do this is
_{4}C_{1}. - Choose one suit for the fifth card in the hand. There are four suits, from which we choose one.
The number of ways to do this is
_{4}C_{1}.

The number of ways to produce a straight (Num

_{s}) is equal to the product of the number of ways to make each independent choice. Therefore,Num

_{s}=_{10}C_{1}*_{4}C_{1}*_{4}C_{1}*_{4}C_{1}*_{4}C_{1}*_{4}C_{1}Num

Conclusion: There are 10,240 different poker hands that can be classified as a straight - either a straight flush or an ordinary straight._{s}= 10 * 4 * 4 * 4 * 4 * 4 = 10,240 - 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
- Finally, compute the probability of being dealt a straight. There are 2,598,960 unique poker hands. Of those, 10,240 are some form of straight. Therefore, the probability
of being dealt a straight (P
_{s}) is:P

_{s}= 10,240 / 2,598,960 = 0.003940037553

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

P_{os} = P_{s} - P_{sf}

P_{os} = 0.003940037553 - 0.00001539077169

P_{os} = 0.003924646781

where P_{s} is the probability of any type of straight, P_{sf} is the probability of a straight flush, and P_{os} is the
probability of an ordinary straight.

Bottom line: In stud poker, even an ordinary straight is a pretty rare event. On average, it occurs once every 255 deals.

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