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I'm not sure how to approach this problem from homework. The only equation I have for calculating probability is if each outcome in the sample space has an equal opportunity. The below isn't equal opportunity right?

A drawer contains 6 red socks, 4 green socks, and 2 black socks. Two socks are chosen at random. What is the probability that they match?

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Please add the homework tag to your question. Hint: you have to count to solve this problem. How many different sets of two socks can be formed from 12 socks? That tells you the size of the sample space: the totality of the possible outcomes that you might observe, and the assumption, deducible from the phrase "chosen at random" means that these outcomes are equally likely to occur. Then count how many pairs of red socks are possible, how many pairs of green socks and how many pairs of black socks could be formed. You might want to ponder the meaning of $\binom{n}{k}$ before starting. –  Dilip Sarwate Jan 24 at 19:44
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I like the generalization of this problem: when you choose four socks at random, what is the chance that you can select a matched pair from them? :-) –  whuber Jan 24 at 20:13

2 Answers 2

Here are all the different ways of pairing one sock with another different one, arranged systematically:

Figure

"Random" in this context means that each pairing is equally likely. Therefore, to find the probability of a color match, count the color-matched pairs and divide by the total.

Counting is made easier by noting that the color-matched socks occur within squares of $6$ by $6$, $4$ by $4$, and $2$ by $2$, and indeed the full tableau of pairings is a square of $12=6+4+2$ by $12$. Within each square the diagonal is missing. Thus, the number of pairs within any such square-without-diagonal of sides $k$ must be $k\times k - k = k(k-1)$. Sum these values over the red socks ($k=6$), green socks ($k=4$), and blue socks ($k=2$), finally dividing by the total ($k=12$).

A general formula for any numbers of various kinds of socks follows immediately.

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Try starting here: What is the probability of drawing two red socks? Well, there are 6 red socks, and you need to choose two of them, meaning there are 6C2 (read "6 choose 2") ways to succeed, or 15 ways to draw two red socks. There are 12 total socks, and you choose 2, so there are 12C2 total combinations of 2 socks, or 66 total combinations of two. Since you have 66 possible combinations, and 15 of them are ways to succeed in drawing two red socks, the probability of drawing two reds is 15/66 or 22.7%. Hopefully you can take it from there.

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