Dice Game with Stopping 2

You will roll a fair die until the game stops. The game stops when you get a 4, 5, or 6. For every number 1, 2, or 3 you have thrown, your score increases by +1. If the game stops with a 4 or 5, you get paid the accumulated score. If the game stops with a 6, you get nothing. What is the expected payoff of this game?

Answer

The recursion equation for the expected payoff $$E$$ is:

E = \tfrac{1}{2} \times 0 \;+\; \tfrac{1}{2} \times \Big(\tfrac{2}{3} + E\Big).

Note that we use $\tfrac{2}{3} + E$ instead of $$1 + E$$ because the score of 1 is only realized with probability $\tfrac{2}{3}$ , on the next throw, when a 4, or 5 is rolled.

Solving the equation $E = \frac{1}{2}(0) + \frac{1}{2}\Big(\frac{2}{3} + E\Big)$ gives:

E = \frac{2}{3}.