Drawer contains 14 socks. If I randomly pull 2, what is the probability that I will get a matching pair?

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.

• 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? – Adriano Aug 30 '13 at 1:27
• P(1st sock is blue) = 2/7 P(1st 2 socks being blue) = ?? – DHShah01 Aug 30 '13 at 1:28
• Can you explain how you got the $2/7$? – Adriano Aug 30 '13 at 1:30
• What have you tried? We will be able to help you more if we know where you are getting stuck. – 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 $\frac{30}{91} \approx 32.967\%$

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

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

protected by Community♦Mar 20 at 20:55

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).

Would you like to answer one of these unanswered questions instead?