Telescope Optics

You are testing telescope optics before shipping them. The probability of an accurate test result is 95%, and 0.5% of the lenses are flawed. How good is your test?

We define

A = the event that the component fails the test
B = the event that the component is flawed
Ac = the event that the component passes the test
Bc = the event that the component is good

and recall that P[A|B] is defined as the probability of A, given B. Our conditions are that

P[A|B] = P[Ac|Bc] = 0.95

P[B] = 0.005

We can use Bayes' Theorem to determine the conditional probability (but you should be able to see that this is the correct ratio to consider, from basic principles).

Putting in the conditions of our case,

This tells us that only 8.7% of the flagged optics are actually flawed, which is a very high false positive rate.