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: 25% when both trucks are overloaded, 7% when one truck is overloaded, and 0.3% when neither truck is overloaded. The fraction of trucks that are overloaded is 12%. You can assume that the overloading of any two trucks has no relationship.

A) What is the probability of damage while two trucks are on the bridge? Enter your answer to four decimal places (0.XXXX).

B) Continuing with the question above, 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

A).

Let us suppose that 1 = Overloaded and 0 = Not overloaded then we have,

11 = both of the trucks are overloaded

10 = truck 1 overloaded but not truck

01= truck 1 is not overloaded but truck 2 overloaded

00 = none of the truck are overloaded

it is given that: P(1) = 0.12 then P(0) = 0.88

Since the trucks are statistically independent then

P(Dam | 11) = 0.25

P(Dam | 10) = P(Dam | 01) = 0.07

P(Dam | 00) = 0.003

Then for the 1st part we have to find the total probability simply as below:

TP: P(Dam) = P(Dam | 11) * P(11) + P(Dam | 10) * P(10) + P (Dam | 01) * P(01) + P(Dam | 00) * P(00)

= P(Dam | 11) * P(1) * P(1) + 2* P(Dam | 10) * P(1) * P(0) + P(Dam | 00) * P(0) * P(0)

= (0.25 * 0.12 * 0.12) + 2 * (0.07 * 0.12 * 0.88) + (0.003 * 0.88 * 0.88)

= 0.0207 or we can say that (2.07 %)

Hence the probability of damage while two trucks are on the bridge is 0.0207.

B).

Here first we determine the probability that the damage was not caused by overloaded trucks and simply subtract that probability from 1.

P(00 | Dam) = ( P(Dam | 00) * P(00) ) / Total Probability

P(00 | Dam) = ( P(Dam | 00) * P(0) * P(0) ) / Total Probability

P(00 | Dam) = (0.003 * 0.88 * 0.88) / 0.0207

P(00 | Dam) = 0.1122

therefore,

P(damage was caused by one or more overloaded trucks given that bridge was damaged)

= 1 - P(00 | Dam)

= 1 - 0.1122

= 0.8878

answer: 0.8878

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