Suppose you need to drive from point $A$ to $B$, but must pass through point $C$ (one hour from point $A$) along the way. At point $C$ there are three roads you can take, $x$, $y$, and $z$ (and you take one at random each time). $x$ will take you to $B$ in one hour, $y$ will circle you back to $A$ in half an hour, and $z$ will circle you back to $A$ in $1/3$ of an hour (that is, if you take $y$ or $z$, you start over). What is the expected time, $X$, for you to reach $B$, from $A$? My solution is as follows: $$ E[X]=1+\frac13(1)+\frac13(E[X]+1.5)+\frac13\left(E[X]+\frac43\right)\implies E[X]=\frac{41}6 $$
My question is, is my solution correct? If it is, can someone provide me with a strictly algebraic solution (that is, write it in terms of infinite sums) rather than a "renewal" method solution.
