You are allowed to pick the balls one at a time from any bin you like until you find the winning one. Once a ball is picked, it is removed from the game.

(a) What is P (Bj |bc) in English? Use Bayes' rule to find P (Bj |bc). The answer should be different for j = i and j = i. Can you interpret the result?
(b) This is a general equality: P (X, Y |Z ) = P (X |Z )P (Y |X, Z ) (we have used it before). Prove this equality.
(c) Find P (Bj , bj,2 |bc), for j = i and j = i. This is the probability of selecting the winning ball from bin j after the removal of a wrong ball from bin i. Use part (b) to compute this probability. Based on your result, suggest a search algorithm for the winning ball