Hey, this is Presh Talwalkar. This problem was part of a written interview test for Amazon in India. A drawer contains twelve identical black socks and twelve identical white socks. If you pick two socks at random: what is the probability of getting a matching pair? Can you figure it out? Give this problem a try and when you’re ready keep watching the video for the solution. Many people think when you pick two socks they will either match or they will not match. Since the number of black and white socks are equal, the probability of matching should be 50%. But this tempting answer is wrong! The probability of making a pair is actually slightly less than 50%. Let’s calculate and figure out why? There are 12 white socks and 12 black socks. By symmetry the probability of picking a black pair of socks equals the probability of picking a white pair of socks. So let’s calculate the probability of picking a black pair of socks. There is a 12 over 24 chance the first sock is black. This is because there are 12 black socks out of 24 socks total. 12 over 24 simplifies to be 1/2. Now what’s the chance of picking a black sock on the second draw? For the second sock the chance will be 11 out of 23, because there are 11 black socks remaining out of a total of 23 socks remaining. The asymmetry is because we are sampling without replacement. So what does this all mean? It means, the probability of picking a black pair is 1/2 times 11 over 23. This will also be the probability of picking a white pair by symmetry. So, we can put this all together to get the odds of picking a pair of matching socks. The probability of picking a matching pair will equal the probability of picking a black pair plus the probability of picking a white pair. Each of these is equal to 1/2 times 11 over 23. And therefore the probability of picking a pair of matching socks will be 11 over 23, which is approximately 47.8%. This is slightly less than 1/2. Did you figure it out? Thanks for watching this video. Please subscribe to my channel. I make videos on math. You can catch me on my blog mindyourdecisions. If you like this video you can support me on patreon… …and you can check out my books which are listed in the video description. You can also follow me on social media either at mindyourdecisions or at preshtalwalkar depending on the site.