Statistics: A Sock Problem

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Problem: Given 6 pairs of socks (a.k.a a total of 12 socks) , where 3 pairs are blue and 3 pairs are black (a.k.a total number of blue socks is 6 and total number of black socks is 6). What is the probability of picking a matching pair ?

Answer: 5/11

Let’s first understand the question: What is the probability of picking a matching pair ? This is just asking us the number of ways to pick any two socks that are the same color.

Initially we are given 6 pairs of socks for a total of 12 socks. 3 pairs are black and 3 pairs are blue, meaning 6 socks are black and the other 6 are blue. We choose the first sock, it doesn’t matter what color the sock is that we choose as long as the second sock is the same color as the first.

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6 Black Socks & 6 Blue Socks

WARNING MATH AHEAD !!

Ways to pick any 2 socks from 12 socks = 12 choose 2 or 12C2.
Ways to pick 2 BLACK socks from 6 BLACK socks = 6 choose 2 or 6C2
Ways to pick 2 BLUE socks from 6 BLUE socks = 6 choose 2 or 6C2

Using the Combination Formula:

n! / k!(n-k)!

12C2 = 12! / 2!(12–2)! = 12!/2!10!= 12 * 11 / 2 = 66
6C2 = 6!/2!(6–2)! = 6!/2!4! = 6*5/ 2 = 15

Probability of picking 2 BLACK socks (P1)= 6C2 / 12C2 = 15/66
Probability of picking 2 BLUE socks (P2)= 6C2 / 12C2 = 15/66

Another way of figuring that out without using the Combination Formula:
Probability of picking 2 BLACK socks (P1)= 6/12 * 5/11=15/66
Probability of picking 2 BLUE socks (P1)= 6/12 * 5/11=15/66

I got the 6/12, from the 6 black socks out of 12 possible socks, and got the 5/11 from choosing the first black sock leaving us with 5 black socks and a total of 11 socks left. The same thing goes for the blue socks

The probability of picking either 2 blue socks or 2 black socks =
Probability of picking any 2 same color socks = P1+P2 = 15/66 + 15/66 = 30/66 = 5/11

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