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A bridge has to be designed for a region which is subject to the influence of both floods and earthquakes. Its expected design life is 120 years. At the particular site, the probability of a flood occurring during any given period of one minute is 10-4 and the probability of an earthquake during any single minute is 10-8. It may be assumed that occurrences of floods and earthquakes are unrelated.

a. In any period of one minute, what is the probability that there will be either an earthquake or a flood or both?

b. If the events in successive minutes are statistically independent, what is the probability that there will be no flood for any period of one month? Or for a period of one year?

c. What is the probability that there will be no earthquake and no flood for any period of one year?

d. Is it sensible that the bridge should be designed for the simultaneous occurrence of floods and earthquakes? Justify your answer.

Okay I am posting this question to make sure I am correct.Just for cross reference

For a) I did P(FUE) = P(F)+P(E)-P(FNE)=(10^-4)+(10^-8)-(10^-4X10^-8) = 10^-4

  Not sure if i am correct about this

For b)The probability no flood, P(F^) = 1-10^-4 = 0.9999 So for 1 month it will be 0.9999^(60x24x30) = 0.01333 So for 1 year it will be 0.9999^(60x24x365)

For c) Probability no flood or earthquake, P(F^NE^)=(1-10^-4)x(1-10^-8) = 0.9998999. So in one year would be same procedure as for b).

For d) This is one part that I am still confused about. I guess you could do the same as probability for both earthquake and flood occuring in the 120 years bridge life and find out whether the probability is high or low which in turn could determine whether to design the bridge in such a way.

Please have a read through if you have time, and let me know about any ideas.

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  • $\begingroup$ An earthquake might result in a flood. Are there any dams upstream from the bridge? $\endgroup$ Apr 17 '16 at 16:58
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    $\begingroup$ @DilipSarwate in the real world, that would be very good advice...I think this is a practice question though. The preamble says to assume floods and earthquakes are unrelated. $\endgroup$
    – user75138
    Apr 17 '16 at 17:43
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    $\begingroup$ Yeah, Fukushima anyone? Anyhow, is that really correct of 1e-4 probability of flood each minute, independent across time? Man, that bridge is going to get a heck of a lot of flooding. $\endgroup$ Apr 17 '16 at 18:35
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Since this is self-study, I'll give hints where I think they will help:

a) Correct approach. Since you are dealing with events of vastly different probabilities, your approximation of $10^{-4}$ is numerically, as long as we're just talking probabilities.

b) Approach looks good here too.

c) You could have calculated this as one minus the probability you got in (a). Then apply the method you used in (b). From De Morgan's Law:

$$P((F\cup E)^c)=P(\neg F \cap \neg E)$$

d) This is an odd question, since no costs are provided. If earthquakes or floods by themselves will do only slight damage, but combined would lead to collapse, then it might be worthwhile.

However, I think your line of thinking here is correct. How likely is such an event over the life of the bridge? I think this is what they are after.

A more nuanced analysis would require additional assumptions about costs. For example, assume that the losses would be large(say 10 Billion in direct repairs and liability/damages/lawsuits). Now, assume that the extra reinforcement would cost $500 Million. Can a case be made that its still not worthwhile?

For an added twist, you can calculated the expected next present value of avoided losses against the extra up-front costs (say, at 6% discount rate)...what do you get then?....see, this can get complicated if you think too hard (i.e., overthink it like this);-)

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  • $\begingroup$ Thanks Bey for the input. So for c, is it the correct method that I used ? Since its the probability of both events not occuring in one year period. Plus for a) so adding them and taking away the intersection of both is correct yes ? $\endgroup$ Apr 17 '16 at 22:35
  • $\begingroup$ One more question Bey, would there be a need to take away the intersection of the earthquake and flood for part a). Would it be simply easy to add probability of earthquake and flood ? $\endgroup$ Apr 17 '16 at 23:35
  • $\begingroup$ @ZaffarMahmood your original calculations were correct. For $P(E \cup F)$, you cannot ignore the joint probability, however small, if you want to be theoretically correct. Practically speaking, it would matter little. $\endgroup$
    – user75138
    Apr 18 '16 at 3:25

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