A very innocent monkey throws a fair die. The monkey will eat as many bananas as are shown on the die, from 1 to 5. But if the die shows '6', the monkey will eat 5 bananas and throw the die again. This may continue indefinitely. What is the expected number of bananas the monkey will eat? Hint Answer
This uses law of total probability very silently, i.e., E( E(X|Y) ) = E(X). Suppose X expected number bananas eaten overall and Y = number shown on dice.
E1 = E( X | Y = 1,2,3,4,5) = 3
E2 = E( X | Y = 6) = 5 + E(X)
now, E(X) = 5/6*E1 + 1/6*E2
Solving this simple equation gives E(X) = 4