You roll a single die. For each roll, you are paid the face value. If a roll gives 1, 2, or 3, you can roll the die again. Once you get 4, 5, or 6, the game stops. What is the expected payoff of this game?
E[X] is the expected payoff of this game. Then we have \begin{equation}
E[X] = \frac{1}{2}*E[\{1,2,3\}] + \frac{1}{2} * E[\{4,5,6\}]
\end{equation} \begin{equation}
E[X] = \frac{1}{2}*(2+E[X]) + \frac{1}{2} * 5
\end{equation} \begin{equation}
E[X] = 1 + \frac{1}{2} E[X] + 2.5
\end{equation} \begin{equation}
E[X] = \frac{1}{2} E[X] + 3.5
\end{equation} \begin{equation}
\frac{1}{2} E[X] = 3.5
\end{equation}Multiplying both sides by 2 gives us \begin{equation}
E[X] = 7
\end{equation}
E[X] = \frac{1}{2}*E[\{1,2,3\}] + \frac{1}{2} * E[\{4,5,6\}]
\end{equation} \begin{equation}
E[X] = \frac{1}{2}*(2+E[X]) + \frac{1}{2} * 5
\end{equation} \begin{equation}
E[X] = 1 + \frac{1}{2} E[X] + 2.5
\end{equation} \begin{equation}
E[X] = \frac{1}{2} E[X] + 3.5
\end{equation} \begin{equation}
\frac{1}{2} E[X] = 3.5
\end{equation}Multiplying both sides by 2 gives us \begin{equation}
E[X] = 7
\end{equation}
Correct Answer: 7
My Notes
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