A monkey reached a banana farm. As a rational monkey, it wants to let a dice decide the number of bananas to eat. The rules of the game are:
If the die finds 1 to 5, it eats that many bananas
If the die gets 6, it eats 5, tosses again, and the game continues.
What is the expected number of bananas that the monkey eats?
The exepcted value is: (1/6) x 1 + (1/6) x 2 + (1/6) x 3 + (1/6) x 4 + (1/6) x 5 + (1/6) x [5 + (1/6) x 1 + (1/6) x 2 + …]
= [20/6] + (1/6)[20/6] + (1/62)[20/6] + …
= [20/6] x [1 + 1/6 + 1/62 + 1/63 + …]
= [20/6] x [1/(1-1/6)]
= [20/6] x [6/5] = 20/5 = 4
Note we used the relationship for the infinite series 1 + x + x2 + x3 + … = 1/(1-x).
