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This question has been answered previously but it didn't explain the logic in plain English. Can someone please go step by step and explain what it is being calculated and how (using numbers would be very helpful, i.e. you pick $2$ socks from $3$ pairs of socks and you get this, that, etc.)?

The question: Suppose you have $n$ pairs of socks in a drawer and you pick out $k$ socks. What is the average number of pairs of socks that you will have if you repeat this experiment a large number of times?

123
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    Previous answer: https://math.stackexchange.com/questions/3370766/expected-value-of-sock-pairs-from-k-choices – 123 Oct 01 '20 at 12:26

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This is a straight forward exercise in Linearity of Expectation.

There are $\binom {2n-2}{k-2}$ ways to choose $k$ socks such that a given pair is selected. There are $\binom {2n}k$ unrestricted ways to choose $k$ socks. Thus the probability that a given pair is selected is the ratio: $$\binom {2n-2}{k-2}\Big / \binom {2n}k=\frac {k(k-1)}{(2n)(2n-1)}$$

Altertnatively: the probability that the left sock is selected from the $k$ is $\frac k{2n}$ and then the probability that the right sock is selected given that the left one was selected is $\frac {k-1}{2n-1}$ so the probability that both are selected is $$\frac k{2n}\times \frac {k-1}{2n-1}$$ which is, of course, equivalent to the prior result.

As there are $n$ pairs the answer is $$\boxed {\frac {k(k-1)}{2(2n-1)}}$$

lulu
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