Bertrand's box, a gold coin appears
Three boxes each hold two coins: one box has two gold coins, one has two silver, and one has one gold and one silver. You pick a box at random, then draw one coin at random from it. It is gold.
What is the probability that the other coin in that same box is also gold?
Show a hint
The tempting answer "it must be one of the two mixed-or-gold boxes, so " ignores that a gold-gold box is twice as likely to have handed you a gold coin. Weight by that likelihood.