Suppose there are five levels $\{1,2,3,4,5\}$. Let $P_s(i)$ denote the probability of success on level $i$ and $P_f(i) = 1 - P_s(i)$ the probability of failure on level $i$. If you succeed on level $i$ you proceed to level $i + 1$; if you fail you go down to level $i - 1$ (unless you are at level $1$, in which case you stay)
I am trying to figure out the expected number of attempts I would need to get from level $4$ to $5$. I know that the result should be some kind of recursion but all I could come up with was
$E_{5} = 1 \cdot P_s(4) + E_{4} \cdot P_f(4)$
where $E_{i+1}$ is the expected number of attempts to get to level $i+1$ from level $i$, and $E_{4} = 1 \cdot P_s(3) + E_{3} \cdot P_f(3)$ and so on.
I know this isn't correct because the expected number of attempts should increase as $P_s(4)$ goes down
Any help on how to figure this out would be appreciated
I take it that the probabilities of success/failure at each step are known (except you can't fail at step $1$)
On this assumption and simplifying notation, denoting the probability of success/failure at each step $i$ as $p_i, (1-p_i)$, and letting $S1, S2,$ etc represent being at step $1,2$ etc, we get the following step by step equations.
$S4= 1 + (1-p_4)(S3) \tag1$
$S3= 1 + p_3(S4) + (1-p_3)(S2) \tag2$
$S2= 1 + p_2(S3) + (1-p_2)(S1) \tag3$
$S1= 1 + p_1(S2) + (1-p_1)(S1) \tag4$
Solve the system of linear equations for $S4$