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**How is this probability reasoning wrong??**

**1. Homework Statement**

There are light socks and dark socks. Francis reasons as follows about socks.

"If I pull out 3 socks, at least 2 will be alike. Suppose that they are dark. The third one will also be dark with probability 1/2. Similarly, if the pair is light, the probability that the remaining one will be light is also 1/2, so the probability of all socks being the same color must be 1/2."

What is wrong with this argument? What is the actual probability of pulling out three socks of the same color.

**2. Homework Equations**

The professor didn't say it explicitly, but I'm pretty sure that sock choices are taken to be independent, so P(A,B,C) = P(A)P(B)P(C).

**3. The Attempt at a Solution**

The last part is easy...the correct probability is P(three socks of the same color) = P(all light) + P(all dark) = 1/2*1/2*1/2 + 1/2*1/2*1/2 = 1/4.

But I CANNOT figure out where Francis's reasoning is wrong. I've been thinking about it for so long... He seems to be completely right, I mean, when you grab 3 socks you ALWAYS end up with 2 matching socks, and the probability of the other sock matching those two is 1/2...right? So from that reasoning the probability of all socks matching is 1/2.

I've thought about it and thought about it and thought about it and I can't figure it out! It's driving me crazy!!! If you can find the flaw I would really appreciate it if you could please explain it or give me some good hints. Thanks to anyone who responds.