Repeating Dice

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}