Consider the communication network shown in the figure below and suppose that each link can fail with probability p. Assume that failures of different links are independent.
Assume that p=1/3. Find the probability that there exists a path from A to B along which no link has failed. (Give a numerical answer.)
- unanswered
Given that exactly one link in the network has failed, find the probability that there exists a path from A to B along which no link has failed. (Give a numerical answer.)
1. 0.46091
2. 0.8
Well, the probability of a single link failing is 1/3. So, the probability of a single link NOT failing is 2/3.
Now, to find the probability that there exists a path from A to B along which no link has failed, we need to consider all possible paths. Since failures of different links are independent, we can just multiply the probabilities of each link NOT failing along the path.
In this case, there are 3 possible paths from A to B:
1. A --> C --> B
2. A --> D --> B
3. A --> E --> B
For each of these paths, the probability of no link failing is (2/3) * (2/3) = 4/9.
Since we want there to exist at least one path with no link failure, we need to consider the complement, which is the probability that there is no such path. So, the probability that there is no path with no link failure is (5/9) * (5/9) * (5/9) = 125/729.
And therefore, the probability that there exists a path from A to B along which no link has failed is 1 - 125/729 = 604/729, which is approximately 0.828, or 82.8%.
Now, let's move on to the second part of the question.
Given that exactly one link in the network has failed, the probability of any particular link failing is still 1/3, but this time we know for certain that one link has failed.
Since there are 5 links in total, and we know that exactly one has failed, there are 5 possible scenarios for which link has failed.
In each scenario, there are still 3 possible paths from A to B (same as before), and the probability of no link failing is still 4/9.
Therefore, the probability that there exists a path from A to B along which no link has failed, given that exactly one link has failed, is (1/5) * 3 * (4/9) = 12/45, which simplifies to 4/15.
So, the answer is 4/15, which is approximately 0.267, or 26.7%.
To find the probability that there exists a path from A to B along which no link has failed, we need to consider two cases:
1. When no link has failed:
In this case, the probability of no link failing is (1-p) for each link in the path from A to B. Since the failures are independent, the probability that there exists a path from A to B along which no link has failed is (1-p)^n, where n is the number of links in the path.
2. When exactly one link has failed:
In this case, we need to consider all possible links that could have failed. Since there are n links in total, there are n different possibilities for exactly one link to fail.
For each of these possibilities, we need to calculate the probability that there exists a path from A to B along which no link has failed, given that exactly one link has failed.
Let's assume link j is the one that has failed. In this case, the probability of link j failing is p, and the probability of all other links not failing is (1-p). Since the failures are independent, the probability that there exists a path from A to B along which no link has failed, given that link j has failed, is (1-p)^(n-1).
To find the overall probability of there existing a path from A to B along which no link has failed, given that exactly one link has failed, we need to sum up the probabilities over all the different possibilities:
P = sum[(1-p)^(n-1)]/n
Now, we can substitute p=1/3 and the appropriate values for n to find the numerical answer.