"N-Year" design magnitudes. A system (e.g., a dam or a dike) is said to be designed for the N-year flood if it has a capacity which will be exceeded only by a flood equal to or greater than the N-year flood. The magnitude of the N-year flood is that which is exceeded with probability 1/N in any given year. Assume that successive annual floods are independent.

(a) What is the probability that exactly one flood equal to or in excess of the "50-year flood" will occur in a 50-year period?

(b) What is the probability that exactly three floods will equal or exceed this 50-year flood in 50 years? (c) What is the probability one or more floods will equal or exceed the 50-year flood in 50 years?

(d) If an agency designs each of 20 independent systems-i.e., systems at widely scattered locations-for its particular 500-year flood, what is the distribution of the number of systems which will fail at least once within the first 50 years after their construction? Assume that

(e) In 1958 the 50-year flood was estimated to be a particular size. In the next 10 years, two floods were observed in excess of that size. If the original estimate was correct, what is the probability of such an observation? Such a rare event may be so unlikely that the engineer prefers to believe (i.e., act as if) his original estimate was wrong. In Chaps. 4 to 6 we shall see that such calculations are an important part of decision making.