Medium | Probability |

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

Law of total probability

Answer

4

Solution

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

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

Source: Self, to show an example of recursive probability

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