Pairing Socks!

Probability Level pending

Two hundred red and two hundred blue socks are in a drawer. You reach in and pull two socks at random. You then reach in and pull out two more socks (without looking at the first pair). You continue this process until you have drawn eighty-three pairs of socks (without looking at any of the other pairs). What is the probability that the eighty-third pair is a matching pair?

The answer is of the form a b \displaystyle \frac{a}{b} where a a and b b are coprime positive integers.

Submit a + b \displaystyle a+b .


The answer is 598.

Aaghaz Mahajan by Aaghaz Mahajan , 19, India

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2 solutions

Mark Hennings
Jul 20, 2019

Since you don't look at the socks you have drawn, it does not matter what they are. The desired probability is simply the probability of choosing a matching pair if we just take two socks out of the drawer, which is ( 200 2 ) + ( 200 2 ) ( 400 2 ) = 199 399 \frac{\binom{200}{2} + \binom{200}{2}}{\binom{400}{2}} \; = \; \frac{199}{399} making the answer 598 \boxed{598} .

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Try #02: You have 200 blue and 200 red socks. Since this is without replacement and you pick one, then there are only 399 socks left and only 199 of the matching colour you need. Hence the chance of getting is 199 399 \frac{199}{399} the first time.

Since you don't know what colour the socks you've picked out are - so probability would suggest that an equal amount has been taken out, so it would be the same for all the other 'picks' as well.

Try #01 (Objection): Surely, since it's your 83rd pair (so there are 117 pairs left) wouldn't the chances be 116 331 \frac{116}{331} instead of [ 199 399 \frac{199}{399} ] because you have 83 pairs and they're just as likely to be equally coloured or biased any one way.

Can anyone explain why it can't be 116 331 \frac{116}{331} please?

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