Question

A small one lane bridge is susceptible to damages from heavy trucks. The bridge has room for only two trucks at most. You will investigate event of possible damage (D) to the bridge when two trucks are driving over bridge simultaneously.

A truck weighing station was set near the bridge to estimate the chances of having overloaded trucks. Around 5% of the trucks were estimated to be overloaded ( i.e. above legal limit). The overloading of trucks can be considered statistically independent.

The likelihood of damage to the bridge is 20% when both trucks are overloaded; the likelihood of damage is 5% if only one of the trucks is overloaded, and the likelihood of damage is 0.1% when both trucks are not overloaded.

To set up your solution, consider the events:

• T₁: the first truck is overloaded
• T₂: the second truck is overloaded

A. What is the probability that the trucks are both overloaded at any given moment?

B. Which of the following concepts studied in class would allow you to calculate the expected probability of damage to the bridge P(D)?

1. The theorem of total probability

2. Conditional probability

3. Bayes Theorem

4. The multiplication Rule

C. Calculate the expected probability of damage to the bridge P(D).

Note: round your answer to only three decimals.

D. New data reveals that the probability of having and overloaded truck is actually 7%; therefore, the expected probability of damage increases to 0.008. All other relationships remain the same. If the bridge is damaged, what is the probability that it was caused by an overloaded truck or trucks?

Hint: you can first calculate the probability that damage was not caused by overloaded truck(s)

Note: round your answer to only two decimals

E. Suppose the local government can allocate money for strengthening the bridge such that the probability of damage will be half of the existing bridge (i.e. P(D) = 0.004). The money could also be used to increase inspection frequency of vehicles such that the fraction of overloaded trucks is decreased from 7% to 4%. The engineer responsible for these measurements believes that strengthening the bridge is the best option to reduce the probability of damage. Is this true or false?

Answer 1