4.1. Example System#

A pre-fabricated bridge design is being considered for river crossings in a remote region of the world, as shown in Figure 1. Cities 1, 2 and 3 are labelled C1, C2 and C3, with bridges labelled B1-B4.

../_images/simple-city.png

Question 1: If the probability of failure for an individual bridge is 0.1 per year, compute the probability that you cannot drive from City 1 to City 2

Question 2: If the probability of failure for an individual bridge is 0.1 per year, compute the probability that you cannot drive from City 2 to City 3.

Question 3: If the probability of failure for an individual bridge is 0.1 per year, compute the probability that you cannot drive from City 1 to City 3

Question 4: Suppose there is a critical facility (e.g., a hospital) in City 3. How many bridges would you need to install between City 2 and 3 to ensure that the probability of someone not reaching City 3 from City 2 is less than \(10^{-4}\)?

Question 5: If you had 1 extra bridge to place, which bridge (i.e., B1, 2, 3 or 4) would you place it next to in order to have the biggest decrease of the probability of not reaching City 3 from City 1?

Question 6: If you only had 3 available bridges, which of the 4 bridges in the figure would you remove first, to cause the smallest reduction in failure probability?

Question 7: What should the probability of each bridge be in order to make the probability of not crossing from C1 to C3 less than 0.01?

Question 8: State whether or not dependence would increase or decrease the probability calculated in question 3, and explain why with 1 or 2 sentences.

Assume each prefabricated bridge is made up of a single road deck and 5 separate frames that are joined together. Each bridge will fail if either the road deck fails, or 1 of the frames fails.

Question 9: If the probability of a frame failure is 0.01, what is the required failure probability of the road deck to make sure the bridge failure probability is no more than 0.1?