- #1
Phys12
- 351
- 42
Homework Statement
Find the probability that a hand of five cards in poker contains four cards of one kind.
Homework Equations
The Attempt at a Solution
Solution given in the book:[/B]
By the product rule, the number of hands of five cards with four cards of one kind is the product of the number of ways to pick one kind, the number of ways to pick the four of this kind out of the four in the deck of this kind, and the number of ways to pick the fifth card.
This is:
C(13, 1)C(4, 4)C(48, 1).
By Example 11 in Section 6.3 there are C(52, 5) different hands of five cards. Hence, the probability that a hand contains four cards of one kind is
C(13, 1)C(4, 4)C(48, 1)
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C(52, 5)
I understand the last part (C(48,1)), we have chosen our 4 cards, now there are 48 cards remaining in the deck and we need the 5th card (choosing only 1 card), hence, we get C(48, 1). However, I don't get the first two (I kind of get C(13, 1), but not C(4,4)).
If I imagine a deck placed in 13 different sections each section containing 4 of each kind, then I get why we will have C(13,1), since of those 13 sections, we need to pick one kind. Is that a correct way of thinking about it? And if yes, then why even mention C(4,4) since we're picking up all the 4 cards when we pick 1 kind?