Suppose there is a system with n components, and we know from past experience that any particular component will fail in a given year with probability p. That is, letting Fi be the event that the i th component fails within one year, we have

for 1 ≤ i ≤n. The system will fail if any one of its components fails. What can we say about the probability that the system will fail within one year? Let F be the event that the system fails within one year. Without any additional assumptions, we can’t get an exact answer for Pr|f|. However, we can give useful upper and lower bounds, namely, 

 

We may as well assume p < 1=n, since the upper bound is trivial otherwise. For example, if n = 100 and p =10-5, we conclude that there is at most one chance in 1000 of system failure within a year and at least one chance in 100,000. Let’s model this situation with the sample space S ::= P({1; : : : ; n})whose outcomes are subsets of positive integers ≤n, where s 2 S corresponds to the indices of exactly those components that fail within one year. For example, {2; 5} is the outcome that the second and fifth components failed within a year and none of the other components failed. So the outcome that the system did not fail corresponds

to the empty set, ;.

(a) Show that the probability that the system fails could be as small as p by describing appropriate probabilities for the outcomes. Make sure to verify that the sum of your outcome probabilities is 1.

(b) Show that the probability that the system fails could actually be as large as n p by describing appropriate probabilities for the outcomes. Make sure to verify that the sum of your outcome probabilities is 1.

(c) Prove inequality (16.11).