Probability of murdering a guinea pig in system of 4 rooms

Question

A guinea pig is in a system of 4 rooms as described in the following illustration (where the dolphin is the guinea pig and skull the choking poison):

The guinea pig can move right in probability $p$ and left in probability $q=1-p$. In room 1 the guinea pig is chocked, dies and can't move anymore. After breaking out of the system (See 'Outside'),it won't come back. Find probability for its death.

My problem is there are infine tracks which all of them include the step $2\to1$.If I understood correct for each track I need to calculate the probability and sum it (e.g $A=3\to 4\to 3\to2\to1$ has probability $P(A)=(1-p)^3p$) but I can't find any common motive except the trivial step toward the trap. How can I sum all the tracks?

Answer 1

Let $P_i$ be the probability of survival if the guinea pig starts in room $i$. Clearly, $P_1=0$ and $P_5=1$ (with room $5$ denoting the outside) and for the intermediate rooms we have $P_i = pP_{i+1}+(1-p)P_{i-1}$ because the little fellow survies iff it survives the same game starting one move later, i.e. either one room to the right or to the left with appropriate probabilities. This gives you three linear equations in the three unknowns $P_2, P_3, P_4$.