I am struggling to represent a situation using the correct mathematical notation. I have a graph that consists of a series of nodes (1-4) and edges which represent movement pathways between nodes. The edges have an attribute d, which represents the distance between the two adjacent nodes.
The probability of successfully moving between two adjacent nodes i and j is a function of distance (d):
$$P_{success} = e^{-d_{i,j}/10}$$
e.g. for each adjacent pair of nodes in the figure above the probability of success would be:
$$P_{success, 1,2} = e^{-0.5/10} = 0.95$$
$$P_{success, 2,3} = e^{-0.1/10} = 0.9$$
$$P_{success, 3,4} = e^{-1.5/10} = 0.86$$
Now I apply an additional toll to the probability of success ($P_{toll}$) which can range from 0 to 1. If $P_{toll}$ = 0.5 then the probability of moving from node 1 to node 4 would be:
$$(0.95 * 0.5) * (0.9 * 0.5) * (0.86 * 0.5) = 0.09$$
My Question:
How do I present a general form of the above situation mathematically?
This is my attempt (which I know is wrong), hopefully I am at least on the right track…
$$\prod^j_{i:j}(e^{-d_{i,j}/10} P_{toll})$$
Edit
To clarify, I am not only interested in the distance from node 1 to 4, but from any node to any other node.