Random walk

All possible outcomes can be represented by a tree diagram The numbers indicates the probabilitiy of Paul reaching the corresponding state. Paul gets to his front door on the first try with a probability $p_1 = \frac{1}{4}$ , on the second try with $p_2 = \frac{1}{16}$ , and so on. For the $m$ th try, the probability reads $\begin{aligned} p_m &= \frac{1}{4^m} \\ \Rightarrow \quad P &= \sum_{m = 1}^\infty p_m = \frac{1}{4}\sum_{m = 0}^\infty \frac{1}{4^m} = \frac{1}{4} \frac{1}{1 - 1/4} = \frac{1}{3} \end{aligned}$ with the help of the geometric series $\sum_{m= 0}^\infty r^m = \frac{1}{1 - r}$ .