100 socks

Here's a pretty easy question to solve in your spare time. Let's say you have 100 colored socks and four colored bags. The socks and bags are colored in various (and different) colors: You have to match socks and bags that's in the same color. (For example, matching a red sock with a red bag is a correct pair) What's the probability that you would manage to match only 99 pairs of socks and bags correctly?

Hint: The answer might be more simple than you thought.

40% 0% 39.1% 20%

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1 solution

Roy K
Feb 25, 2021

Here's the oversimplified answer in case you want to check your answer fast: Since you matched 99 socks correctly, the other 1 pair has to be correct. So the answer is... [Drum Roll] 0%.

It's all pretty simple when you reduce the number of the socks. Let's say we have 3 socks and bags in red, blue and orange. If you match the blue bags with blue socks, and the red bags with red socks, there's no possibility of mismatching the rest. The result is same on a big scale, such as 150 socks or even 10,000 socks.

Conclusion: It's impossible to correctly match only N 1 N-1 socks out of N N socks in this kind of situations.

The reason I answered 0% was because the question stated you have "100 socks", so "99 pairs" would require 198 socks, making it an impossible scenario.

Tristan Goodman - 3 months, 2 weeks ago

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Oh... It looks like I made a mistake! I'll fix it now.

Roy K - 3 months, 2 weeks ago

Haha, same

Veselin Dimov - 3 months, 2 weeks ago

To follow on what Tristan said, you need to say "100 pairs of socks" initially. Or by pair do you mean a pairing between sock and bag?

A more general issue is that you are presenting this as a problem of probability, but I'm not seeing any probability distribution mentioned here.

In any case, your conclusion is flawed. How are you defining a correct pairing? If it's sock to bag, then it's entirely possible to misplace exactly one sock.

I went for 0% based on the idea that the socks come in pairs and it would make no sense to put a pair of socks into two different bags. But there is a real issue with asking for a probability as a solution when no actual randomness is apparently being used. In that context, 0% is the only possible answer, since it can also mean "impossible". With no random distribution specified, how could one end up with 20% or 40%?

Richard Desper - 3 months, 2 weeks ago

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