Second blue draw from a randomly chosen bowl

There are three bowls: one holds two blue balls, one holds two red balls, and one holds one ball of each colour. You choose a bowl uniformly at random and draw a ball, which turns out to be blue. You put it back and draw again from that same bowl. What is the probability that the second ball is also blue?

$\tfrac56$. The first draw is blue with probability $\tfrac13(1+\tfrac12+0)=\tfrac12$, and two blues (with replacement) occur with probability $\tfrac13(1+\tfrac14+0)=\tfrac{5}{12}$. Dividing gives $\tfrac{5/12}{1/2}=\tfrac56$; the blue draw tilts you towards the all-blue bowl.