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If I drew 9 cards out of a standard deck, without replacement, what is the probability of getting a nine card straight as well as a nine card straight flush?

Please represent your answer as a combinatoric expression.

User Mimiz
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1 Answer

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Final answer:

The probability of getting a nine card straight from a standard deck of 52 cards is 4 divided by the number of combinations of choosing 9 cards from 52.

The probability of getting a nine card straight flush is the combinatoric expression:
P(Straight Flush) = 4 / (52 choose 9).

Step-by-step explanation:

To calculate the probability of getting a nine card straight without replacement from a standard deck of 52 cards, you first need to recognize that there are four possible starting points for a straight (because a straight can start with an Ace, 2, 3, or 10). For each straight, there are 9! ways to arrange the cards, but as order does not matter, we should not consider these permutations. So, the number of ways to get a straight is simply 4 (as there are four suits and thus four possible straights).

The probability of getting a specific nine card straight sequence is the combinatoric expression:
P(Straight) = 4 / (52 choose 9), where "52 choose 9" represents the total number of ways to choose 9 cards from 52 without regard to order.

For a nine card straight flush (which is a straight all in the same suit), there are only four possibilities (one for each suit). The probability of getting a nine card straight flush is the combinatoric expression:
P(Straight Flush) = 4 / (52 choose 9).

User Satyam Koyani
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