The network below represents a network of (rather unstable) rope bridges between six villages in some remote corner of Mangrovia. The bridges are so narrow that travel is only possible in one direction (as indicated by the arrows). Further the bridges are now so worn that there is an uncomfortable high chance of them collapsing when used. Assume that the edge labels are the likelihood (as percentage) of a bridge collapsing when the crossing is attempted. So for example there is a 20% chance of the bridge from vertex 2 to 5 collapsing, or a probability of 0.8 of gettig across unharmed.
The network below represents a network of (rather unstable) rope bridges between six villages in some remote corner of Mangrovia. The bridges are so narrow that travel is only possible in one direction (as indicated by the arrows). Further the bridges are now so worn that there is an uncomfortable high chance of them collapsing when used. Assume that the edge labels are the likelihood (as percentage) of a bridge collapsing when the crossing is attempted. So for example there is a 20% chance of the bridge from vertex 2 to 5 collapsing, or a probability of 0.8 of gettig across unharmed.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
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Transcribed Image Text:The network below represents a network of (rather unstable)
rope bridges between six villages in some remote corner of
Mangrovia. The bridges are so narrow that travel is only
possible in one direction (as indicated by the arrows). Further
the bridges are now so worn that there is an uncomfortable
high chance of them collapsing when used.
Assume that the edge labels are the likelihood (as percentage)
of a bridge collapsing when the crossing is attempted. So for
example there is a 20% chance of the bridge from vertex 2 to 5
collapsing, or a probability of 0.8 of getting across unharmed.
30
2
20
20
42
6.
38
20.
25
30
20
20
![The aim is to find a path from vertex 1 to 6 which maximizes the probability
of getting to vertex 6 alive.
[Do all calculations of probabilities below to 4 decimal places]
1. Find the probability of survival for each leg in the network and draw the
network with these probabilities shown.
2. What is the probability of surviving a journey from 1 to 6 along legs (1,2),
(2,5) and (5,6)?
3. Prove that for this problem the Principle of Optimality holds: (i.e. "The
optimal path from any node to the end does not depend on how you arrived
at that node"
adapt the three line proof given in the lecture for the
shortest/longest path to this case) and de
problem.
a DP recurrence for for this
4. Showing your working on the diagram, solve the problem by DP to find the
safest route from 1 to 6.
5. Repeat the calculation in a table form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e09be7d-8a95-4622-860b-6e5f40e00af4%2F246a64e5-ef40-453f-ac38-1a83b76c815d%2Ff4cgzjl_processed.png&w=3840&q=75)
Transcribed Image Text:The aim is to find a path from vertex 1 to 6 which maximizes the probability
of getting to vertex 6 alive.
[Do all calculations of probabilities below to 4 decimal places]
1. Find the probability of survival for each leg in the network and draw the
network with these probabilities shown.
2. What is the probability of surviving a journey from 1 to 6 along legs (1,2),
(2,5) and (5,6)?
3. Prove that for this problem the Principle of Optimality holds: (i.e. "The
optimal path from any node to the end does not depend on how you arrived
at that node"
adapt the three line proof given in the lecture for the
shortest/longest path to this case) and de
problem.
a DP recurrence for for this
4. Showing your working on the diagram, solve the problem by DP to find the
safest route from 1 to 6.
5. Repeat the calculation in a table form.
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