Probability of two vertices being connected by some path in a random directed graph
Consider a BFS exploration process, which proceeds in $k$ stages. Put $V_0 = \{u\}$. Given $V_0,\ldots,V_i$, explore all edges from $V_i$ to $V \setminus \bigcup_{j=0}^i V_j$ (where $V$ is the set of all vertices), and set $V_{i+1}$ to consist of all vertices reached in this fashion; their number has a binomial distribution which can easily be calculated. After $k$ steps, check whether the vertex $v$ belongs to $\bigcup_{j=0}^k V_j$.

Note that this process is exactly the same in both the undirected and the directed case. Hence the answer, whatever it is, is identical for both models. Presumably in the undirected case the answer is known and can be looked up. Otherwise you can try to estimate it by estimating the sizes $|V_i|$ and so the probability $\frac{1}{n-1} \sum_{i=1}^k |V_i|$ that $v$ belongs to $\bigcup_{i=1}^k V_i$.