Calculate the number of paths found on the basis of probabilities Let G=<V,E> - weighted directed graph
$w(e_{ij})$ - transition probability from node $v_i$ to node $v_j$ ($w \in[0;1]$)
So my first question is: how calculate the probability of path $u \rightarrow v $ ?
My answer (that I would like to check on the truth): $p(u \rightarrow v) = \prod p(e_j)$, where $e_j$ are edges in this path.  
Second question is: how calculate the probability of reaching from $u\in V$ to $v\in V$ (e.g. using information about the probabilities of transitions or probability of paths $u \rightarrow v $)?
Third question is: Let $P(u,v)$ - probability of reaching from $u$ to $v$. $p(u \rightarrow v)$ - probability of path from $u$ to $v$. Can I count the numbers of paths that still need to be found in addition to the found $u→v$ one?
For example, let:
$P(u,v)=0.5$
$p(u \rightarrow v) = 0.2$ 
$Result = \frac{p(u \rightarrow v)}{P(u,v)}=\frac{0.2}{0.5}*100$% $= 40$% path found.
I'll be glad to answer any of these questions
 A: First question: You are basically right, but let's lay some notation.
I assume your graph is finite (finite edges and finite vertices). Let a path be a set $\{e_i\}$ for $i=1,2,...,k$ where each $e_i$ is an edge and the $e_i$ lands where $e_{i+1}$ starts, for all $i=1,2,...,k-1$.
Now, given this path, call the probability of following it $p\big(\{e_i\}\big)$. This will be equal to the product of the weights of each edge: $$p\big(\{e_i\}\big) =\prod_{i=1}^kw(e_i)$$
Second question: The probability of reaching a vertex $v$ starting from $u$ will be the sum of the probabilities of following a path that starts at $u$ and ends at $v$. Notice that these paths from $u$ to $v$ can in principle include paths that come back to $u$, but the last edge will always be the only one to land at $v$.
Third question: Short answer is no. You can see it from your example: using your method you conclude that you have found $40%$ of the paths from $u$ to $v$. But if $1=\frac{40}{100}N$, then $N=2.5$ which is not an integer!
The reason you cannot do this in general, is that knowing the probability of one path only fixes yo sum of the probabilities of the other paths (in your example this would be $0.5-0.2=0.3$). You could have many other paths, with low probabilities, or just one more path.
In general, to answer the question of how many paths from $u$ to $v$ exist, you will need to look at other properties of the graph.
