Repeating dice

Expected Value
You roll a dice. For each roll, you are paid the face value. If a roll gives 1, 2 or 3, you can roll the dice again. Once you get 4, 5 or 6, the game stops. What is the expected payoff of this game?

Hint

Answer

E[X] is the expected payoff of this game. Then we have

(1)  

(2)  

(3)  

(4)  

(5)  

Multiplying both sides with 2 gives us E[X] = 7.