Let $p_0\in(0,1)$ and let $s>0$. Consider the sequence with terms $p_k$ generated randomly as follows: $$ p_{k+1}=\left\{ \begin{array}{ccl} p_k & \mbox{if} & r_k\leq p_k\\ p_k-s & \mbox{if} & r_k > p_k \end{array} \right., $$ where $r_k\in[0,1]$ is uniformly randomly chosen at each step. We stop whenever $p_k\leq 0$.
What is the expected length of the generated sequence (taking $p_0$ as the first term and the last strictly positive $p_k$ as the last term)?
I can make some simulations and some computations by hand, but I don't really know about any more sophisticated tool to approach such questions. I'm doing this for fun, so any help that might make me learn new stuff will be very welcome.
You can use linearity of expectation. Let $q_0 = p_0, q_1 = p_0-s, q_2 = q_1-s$,... $q_l = q_{l-1}-s$ where we stop so that $q_l-s < 0$. Note this is a pre-determined, deterministic sequence. The expected number of times we see $q_j$ arise in the sequence $(p_0,p_1,\dots,p_k)$ is $\frac{1}{1-q_j}$, so the expected value of $k$ is $\sum_{j=0}^{l-1} \frac{1}{1-q_j}$.