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Chapter 7: Problem 38

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? Four of a kind

Short Answer

Expert verified
The probability of being dealt a "four of a kind" poker hand is \(\frac{1}{4165}\) or approximately 0.00024.

Step by step solution

01

Calculate the number of ways to obtain the 4 cards of the same rank

First, we need to pick one of the 13 card ranks. Then, we have to select all 4 suits of this rank since we need four cards of the same rank. So there are 13 choices for the rank.
02

Calculate the number of ways to obtain the fifth card (odd one out)

Once we have the four cards of the same rank, we are left with 48 cards (52-4) to choose the last remaining card, which must have a different rank. Since there are 12 (13-1) rankings left, there are \(12 * 4 = 48\) combinations available for that last card (12 ranks and 4 suits of each rank).
03

Calculate the total number of ways to obtain a "four of a kind" hand

Now, we can combine Step 1 and Step 2: There are 13 ways to select the rank of the 4 cards and 48 ways to select the last card. These choices are independent, so we can multiply them together to get the total number of "four of a kind" hands: \(13 * 48 = 624\).
04

Calculate the total number of possible 5-card hands from a deck of 52 cards

To find the total number of 5-card hands from a 52-card deck, we can use combination formula (nCr), where n is the total number of cards, and r is the number of cards we want to choose from: \[_{52}C_{5} = \frac{52!}{5!(52-5)!} = \frac{52!}{5!47!} = 2,598,960\]
05

Calculate the probability

Finally, we can calculate the probability of obtaining a "four of a kind" hand by dividing the total number of "four of a kind" hands by the total number of possible 5-card hands: Probability = \(\frac{Number\:of\:Four\:of\:a\:Kind\:Hands}{Total\:Number\:of\:5-card\:Hands}\) Probability = \(\frac{624}{2,598,960}\) = \(\frac{1}{4165}\) Therefore, the probability of being dealt a "four of a kind" poker hand is \(\frac{1}{4165}\) or approximately 0.00024.

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