2.63 The structure shown in the figure could be subject to settlement problems. The likelihood of having a settlement problem {event A] depends on the subsoil condition, in particular, whether a weak zone exists in the subsoil or not. If there is a small weak zone [event 5], the probability of A is 0.2; if the weak zone is large [event L), the probability of A becomes 0.6; last, if no weak zone exists [event N}, then the probability of A is only 0.05. Based on their experiences with the geology of the neighborhood and the soil information from the preliminary site exploration program, the engineers in this case believe that there is a 70% chance of no weak zone in the stratum underlying the structure; however, if there is a weak zone, it would be twice as likely to be small than large. {a} What is the probability that the structure will have a settlement problem? [b] Suppose an additional boring can be performed at the site to gather more information about the presence of the weak material. The engineers judge that: if a large weak zone exists, it is 80% likely that the boring will encounter it; this encounter probability drops to 30% for a small weak zone. Obviously, the boring will not encounter any weak material if the weak zone does not exist at all. Suppose the additional boring will failed to encounter any weak material. i. What is the probability of the presence of a large weak zone? ii. What is the probability of a small weak zone? iii. In this case, what is the probability that the structure will have a settlement