Few Things You Need to Know About Making a Straight Flush
In this article, we will be explaining how to calculate the probability of being dealt a straight flush in stud poker.
What is a Straight?
In stud poker, two types of hands can be classified as a straight.
Straight flush – Five cards with the same suit in sequence, such as 3♥, 4♥, 5♥, 6♥, 7♥.
Ordinary Straight – Five cards in sequence, with at least two cards with different suits. Thus, 10♠, J♥, Q♦, K♣, A♥ and A♠, 2♥, 3♦, 4♣, 5♥ are valid straights.
In this article, we will be calculating the probabilities for a straight flush.
Probability of a Straight Flush
Let us execute the systematic plan to find the chances of a straight flush.
Firstly, count the number of five-card hands that can be dealt from a deck of 52 cards. This becomes a combination problem. n! / r!(n – r)! is the number of combinations. There are 52 cards in the deck, so the value of n is 52. And the cards are dealt in random groups of 5, so the value of r is 5. Thus, the number of combinations becomes:
52C5 = 52! / 5!(52 – 5)! = 52! / 5!47! = 2,598,960
Hence, there are 2,598,960 distinct poker hands.
Next, count the number of ways in which five cards can be dealt to deliver a straight flush. As we have seen earlier, a straight flush consists of five cards in order, each card with the same suit. It needs two independent choices to create a straight flush:
Choose the rank of the lowest card. 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 becomes 10C1.
Choose one suit for the hand. There are four suits, from which we choose one. The number of ways to do this becomes 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: 40 different poker hands 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 chances of being dealt a straight flush is, therefore, 0.00001539077169. On average, a straight flush is dealt one time in every 64,974 deals.