CE 463 Probabilistic and Statistical Methods in Civil Engin eering (S2) July 21, 2016 Topic – Set Theory, Conditional Probability, Bayes Rule Practice Problems
1. Towns A, B and C lie along a river, as shown in figure below, which may be subject to overflow overflow (flooding). The T he annual probability of flooding are 0 .2, 0.3 and 0.1 for towns A, B and C respectively. The events of flooding in each of the towns A, B and C are not statistically independent. If town C is flooded in a given year, probability that town B will also be flooded the same year is increased to 0.6; and if both town B and C are flooded in a given year, the probability that town A will also be flooded that year is increased to 0.8. However, if town C does not experience flooding in given year, the probability that both A and B will also not su ffer any flooding in that year is 0. 9. In a given year, if all three towns are flooded, it is regarded as a disaster year. Suppose the flooding events between b etween any two years are statically independent. Answer the following: (a) What is the probability that a given year in the region is a disaster year? (b) If town B is flooded in a given year, what is the probability that town C is also flooded?
2. The number of accidents at rail-highway grade-crossings reported for a province over the last 10 years are summarized and classified as follows:
Time of occurrence
Day (D) Night (N)
Type of accident (R) Run into Train (S) Struck by Train 30 60 20 20
Suppose there are 1000 rail-highway grade-crossings in province XY. (a) What is the probability that and accident will occur at a given crossing next year? (b) If an accident is reported to have occurred in the d aytime, what is the probability that it is a “struck by train” accident? (c) Suppose that 50% of the “run into train” accidents are fatal and 80% of the “struck by train” accidents are fatal, what is the probability that the next accident will be fatal?
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(d) Suppose D = event that the next accident occurs in the daytime R = event that next accident is a “run into train” accident (i) Are D and R mutually exclusive? Justify (ii) Are D and R statistically independent? Justify
3. An examination of the 10-year record of rainy days for a town reveals the following: 1. 30% of the da ys are rainy days. 2. There is a 50% chance that a rainy day will be followed by another rainy day. 3. There is a 20% chance that two consecutive rainy days will be followed by a third rain y day. A house is scheduled for painting starting next Monday for a p eriod of 3 days. (a) Let E1 = Monday is a rainy day E2 = Tuesday is rainy day E3 = Wednesday is rainy day Express the events corresponding to the three p robabilities indicated above; i.e., 1, 2, and 3 in terms of E 1, E2 and E 3. (b) What is the probability that it will rain on both Monday and Tuesday? (c) What is the probability that Wednesday will be the only dry day during the painting period? (d) What is the probability that there will be at least one rainy day during painting period?
4. A self-standing antenna disk is supported by a lattice structure that is anchored to the ground at the base. During a wind storm, the disk may be damaged as a result of anchorage failure and/or failure of the lattice stru cture. Suppose the following information is known: 1. The probability of anchorage failure during a wind storm is 0.006. 2. If the anchorage should fail, the probability of failure of the lattice structure will be 0.40, whereas the probability of failure of the anchorage given that failure has occurred in the lattice structure is 0.30. Determine the following: (a) The probability of damage to the antenna disk during a wind storm. (b) The probability that only one of the two potential failure modes will occur during a wind storm. (c) If the disk is damaged during a wind storm, what is the probability that it was caused only by anchorage failure?
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