Which shelf given two grey socks drawn

A dresser has a top and a bottom shelf, each holding a huge supply of socks. A sock drawn from the top shelf is grey with probability $0.4$ (white otherwise); one drawn from the bottom shelf is grey with probability $0.7$. You choose a shelf at random with equal odds and draw two socks from it, and both are grey. Draws are independent because the supply is effectively unlimited. What is the probability you chose the top shelf?

$16/65$. By Bayes' rule with equal priors, the posterior is proportional to the likelihoods $0.4^2=0.16$ and $0.7^2=0.49$. So $P(\text{top}\mid GG)=\tfrac{0.16}{0.16+0.49}=\tfrac{16}{65}$.