Network of One-Way Bridges

A flood-prone town is connected by one-way pedestrian bridges arranged in 3 stages. Stage 1 has 2 starting points ($S_1, S_2$), stage 2 has 3 relay islands ($I_1, I_2, I_3$), and stage 3 has 2 destinations ($D_1, D_2$). The bridges are: $S_1 \to I_1$, $S_1 \to I_2$, $S_2 \to I_2$, $S_2 \to I_3$; and $I_1 \to D_1$, $I_2 \to D_1$, $I_2 \to D_2$, $I_3 \to D_2$. A refugee's route is a path $S_i \to I_j \to D_k$ using only existing bridges. If a route is chosen uniformly at random from all valid routes, find $P(\text{the route passes through } I_2 \mid \text{the refugee arrives at } D_1)$.