Question

1. Parts (a) and (b) require the exact same reasoning.

   1. (a) Consider a sequence of Bernoulli trials. if the probability of success is a random variable with uniform distribution between 0 and 1, what is the probability that n trials are needed?

   2. (b) A machine relies on three parts, each of which has a probability 1 − p of failure (and probability p of working). All three parts function independently of the others and the machine is working if and only if at least two of the parts are working. It is easy to see that the probability that the machine is functioning is 3p2(1−p)+p3 (why?). Now suppose that the machine itself is set in a random environment in which the value p becomes random, taking a uniform value bet- ween 0 and 1, independent of the state of the system. Compute the probability that the machine operates in this random environment.

Answer 1

Solution: