A small, old bridge is susceptible to damages from heavy trucks. Suppose the bri

Question

A small, old bridge is susceptible to damages from heavy trucks. Suppose the bridge can have room for at most two trucks, one in each lane. It can easily carry one truck, but it may be damaged when two trucks are present, and this depends on the extent of overloading. The damage probability is: 28% when both trucks are overloaded, 5% when one truck is overloaded, and 0.4% when neither truck is overloaded. The fraction of trucks that are overloaded is 13%. You can assume that possibility of damage when only one truck is on the bridge. What is the probability of damage to the bridge while two trucks are on it? (3 decimal places, 0.XXX) Continuing with the bridge question, if the bridge is damaged, what is the probability that it was caused by one or more overloaded trucks? (3 decimal places, 0.XXX)

Explanation / Answer

5. p(over loaded truck) = 0.13

P(non over loaded truck ) = 1 - 0.13 = 0.87

P(damage) = P(damage dure to 0 overloaded trucks) + P(damage due to 1 overloaded truck) + P(damage due to 2 overloaded trucks)

= 0.87x0.87x0.004 + 2x0.87x0.13x0.05 + 0.13x0.13x0.28

= 0.003 + 0.011 + 0.005

= 0.019

6. p(one or more truck cause damage | damage ) = (0.011+0.005)/0.019

= 0.842

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