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Chapter 7: Problem 36

If a 5-card poker hand is dealt from a well-shuffled deck of 52 cards, what is the probability of being dealt the given hand? A straight (but not a straight flush)

Short Answer

Expert verified
The probability of being dealt a straight (but not a straight flush) in a 5-card poker hand is approximately 0.392%.

Step by step solution

01

Calculate Total Number of 5-Card Poker Hands

For this, we will use the combination formula. A combination is a selection of items from a larger set, such that the order of the items does not matter. The formula for combinations is: \[C(n, k) = \frac{n!}{k!(n-k)!}\] In this case, we have a deck of 52 cards, and we're choosing 5 cards, so the combination is: \[C(52, 5) = \frac{52!}{5!(52-5)!}= \frac{52!}{5!47!}\] Calculating this value, we get: \[C(52, 5) = 2,598,960\] There are 2,598,960 total 5-card poker hands.
02

Calculate Total Number of Straights

A straight consists of 5 consecutive cards that are not all of the same suit. There are 10 possible sets of 5 consecutive cards: A-2-3-4-5, 2-3-4-5-6, 3-4-5-6-7, ..., 10-J-Q-K-A. For each set, we have to multiply by 4 ways each card can have one of four suits. However, we need to deduct the straight flushes (5 consecutive cards with the same suit) as that is not considered in the required probability. There are 4 straight flushes for each of the 10 sets (one for each suit). Total number of straights = (10 sets of consecutive cards) 脳 (4^5 - 4 straight flushes) Total number of straights = 10 脳 (1024 - 4) = 10 脳 1020 = 10,200
03

Calculate Probability of Being Dealt a Straight

Now, we can find the probability of being dealt a straight by dividing the total number of straights by the total number of 5-card poker hands: \[P(straight) = \frac{Total \: number\: of\: straights}{ Total\: number\:of\: 5-card\: poker\: hands }\] \[P(straight) = \frac{10,200}{2,598,960}\] Now, we can simplify the fraction: \[P(straight) \approx \frac{255}{64,974} \approx 0.00392\] The probability of being dealt a straight (but not a straight flush) is approximately 0.392%.

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