What is the probability of randomly choosing ie being dealt 5 cards from a full deck and all of them being spades?

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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

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.

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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.

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,

Ps = Psf + Pos

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

Pos = Ps - Psf

From the analysis in the previous section, we know that the probability of a straight flush (Psf) is 0.00001539077169. Therefore, to compute the probability of an ordinary straight (Pos), we need to find Ps. Here is how to find Ps:

• 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 10C1.
  • 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 4C1.
  • 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 4C1.
  • 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 4C1.
  • 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 4C1.
  • 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 4C1.

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

  Nums = 10C1 * 4C1 * 4C1 * 4C1 * 4C1 * 4C1

  Nums = 10 * 4 * 4 * 4 * 4 * 4 = 10,240

  Conclusion: There are 10,240 different poker hands that can be classified as a straight - either a straight flush or an ordinary straight.
• 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 (Ps) is:

  Ps = 10,240 / 2,598,960 = 0.003940037553

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

Pos = Ps - Psf

Pos = 0.003940037553 - 0.00001539077169

Pos = 0.003924646781

where Ps is the probability of any type of straight, Psf is the probability of a straight flush, and Pos 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.

If you would like to cite this web page, you can use the following text:

Berman H.B., "How to Compute the Probability of a Straight in Stud Poker", [online] Available at: https://stattrek.com/poker/probability-of-straight URL [Accessed Date: 11/6/2022].

If you watch any movie that involves poker, it seems like it’s only a matter of time before a royal flush makes an appearance. This is a poker hand that has a very specific composition: the ten, jack, queen, king and ace, all of the same suit. Typically the hero of the movie is dealt this hand and it is revealed in a dramatic fashion. A royal flush is the highest ranked hand in the card game of poker. Due to the specifications for this hand, it is very difficult to be dealt a royal flush. 

There is a multitude of different ways that poker can be played. For our purposes, we will assume that a player is dealt five cards from a standard 52 card deck. No cards are wild, and the player keeps all of the cards that are dealt to him or her.

To calculate the probability of being dealt a royal flush, we need to know two numbers:

• The total number of possible poker hands
• The total number of ways that a royal flush can be dealt.

Once we know these two numbers, the probability of being dealt a royal flush is a simple calculation. All that we have to do is to divide the second number by the first number.

Some of the techniques of combinatorics, or the study of counting, can be applied to calculate the total number of poker hands. It is important to note that the order in which the cards are dealt to us does not matter. Since the order does not matter, this means that each hand is a combination of five cards from a total of 52. We use the formula for combinations and see that there are a total number of C( 52, 5 ) = 2,598,960 possible distinct hands.

A royal flush is a flush. This means that all of the cards must be of the same suit. There are a number of different kinds of flushes. Unlike most flushes, in a royal flush, the value of all five cards are completely specified. The cards in one's hand must be a ten, jack, queen, king and ace all of the same suit.

For any given suit there is only one combination of cards with these cards. Since there are four suits of hearts, diamonds, clubs, and spades, there are only four possible royal flushes that can be dealt.

We can already tell from the numbers above that a royal flush is unlikely to be dealt. Of the nearly 2.6 million poker hands, only four of them are royal flushes. These nearly 2.6 hands are uniformly distributed. Due to the shuffling of the cards, every one of these hands is equally likely to be dealt to a player.

The probability of being dealt a royal flush is the number of royal flushes divided by the total number of poker hands. We now carry out the division and see that a royal flush is rare indeed. There is only a probability of 4/2,598,960 = 1/649,740 = 0.00015% of being dealt this hand.

Much like very large numbers, a probability that is this small is hard to wrap your head around. One way to put this number in perspective is to ask how long it would take to go through 649,740 poker hands. If you were dealt 20 hands of poker every night of the year, then this would only amount to 7300 hands per year. in 89 years you should only expect to see one royal flush. So this hand is not as common as what the movies might make us believe.