jmc

Tim has 18 total socks, so there are $\left(\genfrac{}{}{0ex}{}{18}{2}\right)=153$ ways for him to pick 2 of them. There are $\left(\genfrac{}{}{0ex}{}{10}{2}\right)=45$ ways for him to pick 2 of his 10 gray-bottomed socks, and $\left(\genfrac{}{}{0ex}{}{8}{2}\right)=28$ ways for him to pick 2 of his 8 white-bottom socks, for a total of $45+28=73$ ways for him to pick a matching pair. So the probability he picks a matching pair is $\boxed{\frac{73}{153}}$.