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Chapter 7: Problem 37

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 flush (but not a straight flush)

Short Answer

Expert verified
The probability of being dealt a flush (but not a straight flush) in a 5-card poker hand is approximately 0.001965 or 0.1965%.

Step by step solution

01

Calculate the total number of 5-card hands

We have a standard 52-card deck and need to choose 5 cards without any order. This can be calculated using combinations: \[C(n, k) = \frac{n!}{k!(n-k)!}\] In our case, n=52 and k=5: \(C(52, 5) = \frac{52!}{5!(52-5)!}\)
02

Calculate the total number of flush hands

There are four suits, and 13 cards in each. We have to choose any five cards of the same suit. And we do this for all four suits. Total flush hands = 4 脳 \(C(13, 5)\)
03

Calculate the number of straight flush hands

A straight flush is a hand in which all five cards are in a consecutive rank sequence and are of the same suit. In each suit, there are ten possible sequences: A-2-3-4-5 to 10-J-Q-K-A. Total straight flush hands = 4 脳 10
04

Calculate the number of flush hands but not straight flush

To find the number of flush hands without counting straight flush hands, subtract the straight flush hands from the total flush hands: Flush hands (not straight flush) = Total flush hands - Straight flush hands
05

Calculate the probability of being dealt a flush (but not a straight flush)

Divide the number of flush hands (not straight flush) by the total number of possible 5-card hands: Probability = \(\frac{\text{Flush hands (not straight flush)}}{\text{Total 5-card hands}}\) Now, we can calculate the values and the probability: Step 1: \(C(52, 5) = \frac{52!}{5!(52-5)!} = 2,598,960\) Step 2: Total flush hands = 4 脳 \(C(13, 5)\) = 4 脳 \(\frac{13!}{5!(13-5)!}\) = 4 脳 1,287 = 5,148 Step 3: Total straight flush hands = 4 脳 10 = 40 Step 4: Flush hands (not straight flush) = 5,148 - 40 = 5,108 Step 5: Probability = \(\frac{5,108}{2,598,960} \approx 0.001965\) The probability of being dealt a flush (but not a straight flush) in a 5-card poker hand is approximately 0.001965 or 0.1965%.

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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)
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