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I begin with a graph $G=(V,E)$. For each edge $e \in E$, I colour the edge with probability $p_e$. I'm looking for the probability that two vertices $v,w \in V$ are the endpoints of a coloured path.

For an example of what I mean, consider the following graph: Coloured graph on six vertices. The path $\left \lbrace (0,5),\, (5,4) \right \rbrace$ has been coloured, and the vertices $\lbrace 0,\, 4 \rbrace$ are endpoints of a coloured path. Note that vertex $5$ is not the endpoint of a coloured path as I define it, since more than one of its incident edges is coloured.

I've taken a look in some basic books about statistics/graph theory (Maurer/Ralston, Grimmett), but not found an easy answer to the problem, or a statement that it's truly difficult. I'm not too familiar with the literature, so reading suggestions would be appreciated.

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    presumably you are looking for an algorithm right? Clearly no simple formula exists. – 2017-01-26
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    @JorgeFernándezHidalgo I'd appreciate an algorithm, an approximation, a special case which permits an algorithm/approximation, or just a suggestion for a textbook to search through. – 2017-01-26
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    I do not know much about the graph theory. Do you mean you want to find the probability that a certain vertex has exactly 1 colored incident edge? If each edge are independent and has the identical probability $p_e$ to be colored, then the number of colored incident edge of the vertex $v$ follows $\text{Binomial}(\deg(v), p_e)$, and under such assumption the probability is $\deg(v)p_e(1-p_e)^{\deg(v)-1}$ – 2017-01-26
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    @BGM I think you may have been misled by my small example. A path on a graph is a set of edges of the form $\lbrace (u_0,u_1), (u_1,u_2), \ldots (u_{n-2}, u_{n-1}) \rbrace$, where $u_0$ and $u_{n-1}$ are the endpoints. You can think of it as a line that "connects the dots". Not only do the edges at the endpoints have to be coloured, those in the middle of the path must also be. – 2017-01-26
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    But if a vertex has exactly 1 colored incident edge, then it must be the endpoint of some path - maybe of those with length 1 only? Do you need include the longest path or have other criterion? – 2017-01-28
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    It's true that if a vertex has one coloured edge, then it's the endpoint of a path, and vice versa. However, I'm looking for the probability that a path exists between two specific vertices, which doesn't always exist if both vertices have a coloured edge incident. – 2017-01-28

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