My drawer contains 4 blue socks, 7 red socks, and 3 yellow socks. If I randomly pull 2 socks at the same time, what is the probability that the socks are the same color?

I know that the probability that the first sock is blue is $\frac4{14}=\frac27 $. But I do not know how to calculate the probability that the first two socks are blue.

  • $\begingroup$ Do you know how to find the probability that the first sock is blue? How about the probability that the first two socks are both blue? $\endgroup$ – Adriano Aug 30 '13 at 1:27
  • $\begingroup$ P(1st sock is blue) = 2/7 P(1st 2 socks being blue) = ?? $\endgroup$ – DHShah01 Aug 30 '13 at 1:28
  • $\begingroup$ Can you explain how you got the $2/7$? $\endgroup$ – Adriano Aug 30 '13 at 1:30
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    $\begingroup$ What have you tried? We will be able to help you more if we know where you are getting stuck. $\endgroup$ – dfeuer Aug 30 '13 at 1:45

You have $14$ socks, so there are ${14 \choose 2} = 91$ ways can you pull $2$ socks out from that pile. Of those $91$ ways, you can get pairs by picking two blues, two reds, or two yellows. There are ${4 \choose 2} = 6$ ways to pick blue socks, ${7 \choose 2} = 21$ ways to pick red socks, and ${3 \choose 2} = 3$ ways to pick yellow socks. So there are $30$ possible "good" outcomes out of $91$ total, so the probability is $\frac{30}{91} \approx 32.967\%$

  • $\begingroup$ Is this question edited later? The solution seems wrong. $\endgroup$ – saint1729 Sep 30 at 14:05
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    $\begingroup$ Yes, @saint1729, checking the history it was edited in 2017 and the 7 red socks were changed to 5. Thanks for the heads up. $\endgroup$ – Avraham Oct 2 at 3:02

As Adriano pointed, we gonna split this problem in 3 sub-problems. What's the probability of drawing a 2 blue socks in the first 2 drawing. It's:

$$\frac{4}{14} \times \frac{3}{13} = \frac{12}{182}$$

Now what's the probability of drawing 2 red socks? It's:

$$\frac{7}{14} \times \frac{6}{13} = \frac{42}{182}$$

And the final sub-problem, what's the probability of drawing 2 yellow socks? It's:

$$\frac{3}{14} \times \frac{2}{13} = \frac{6}{182}$$

Now we add up this 3 fractions and we end up with:

$$\frac{12}{182} + \frac{42}{182} + \frac{6}{182} = \frac{60}{182} \approx 32.97 \%$$


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