I'm a bit of a newbie when it comes to probability theory/terminology, so please excuse any errors on my part. Note that I ask this out of curiosity, so a full answer would be appreciated. Here goes the problem:
Define the probabilistic map $M:\Bbb N\mapsto\Bbb N$ as follows:
$M(n)=\begin{cases}n-1&\text{probability}~p_1\\n&\text{probability}~p_2\\n+1&\text{probability}~p_3\\n+2&\text{probability}~p_4\\n+3&\text{probability}~p_5\\n+4&\text{probability}~p_6\end{cases}$
with $p_1$, ..., $p_6$ constant and $p_1+\cdots+p_6=1$. Now, define $t$ as the first value such that $M^t(1)=0$ (with $M^t$ denoting repetition). For which $p_i$ does the expected value of $t$ exist?
For $p_1=1$, it trivially holds that $M(1)=0$. For $p_1<1$, no $t$ exists such that $M^t(1)=0$ holds with probability 1.