When to replace a maintenance system. An article in the Journal of Quality of Maintenance Engineering (Vol. 19, 2013) studied the problem of finding the optimal replacement policy for a maintenance system. Consider a system that is tested every 12 hours. The test will determine whether there are any flaws in the system. Assume that the probability of no flaw being detected is .85. If a flaw (failure) is detected, the system is repaired. Following the fifth failed test, the system is completely replaced. Now let x represent the number of tests until the system needs to be replaced.

a. Give the probability distribution for x as a formula.

b. Find the probability that the system needs to be replaced after 8 total tests.