Assume that the edge labels are likelihood as percentage of a bridge collapsing when the crossing is attempted shown in the picture below.
Find the Probability of survival for each leg in the network and draw the network with these probabilities shown.
Solve the DP to find the safest route from 1 to 6
My attempted solution is the following where the blue line indicates the safest route from 1 to 6 and the values are probability of moving from node i to node 6
I'm not sure if my working is correct so any insights on how to proceed or start would be very much appreciated
The idea is to compute the highest or Max Probability of Survival reaching to a node n - call it P(n). Call $PS(A,B)$ as Probability of survival from node A to B. So, $$P(6) = Max\{P(4), P(5), P(2), P(3)\}$$ $$P(4) = Max\{P(2), P(3)\}$$ $$P(5) = Max\{P(2), P(3)\}$$ $$P(2) = 0.8, P(3) = 0.75$$ $$P(5) = Max\{PS(2, 5)*P(2), PS(3,5)*P(3)\} = Max\{0.8*0.8, 0.8*0.75\} = 0.64$$ Best path is from 2 to 5.
Similarly $$P(4) = 0.6$$ Path 1->3->4.
So, $$P(6) = Max\{ PS(4,6)* P(4), PS(5,6)*P(5), PS(2,6)*P(2), PS(3,6)*P(3) \}$$ $$P(6) = 0.465$$
So 1->3->6 is the Best path with highest Probability of Survival.