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 the probability of no flaw being detected is .85. If a flaw (failure) is detected, the system is repaired. Following the 5th 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.
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