My real application is probabilistic detection, but it can be conveyed in a simple toy problem (fake numbers as placeholders) ...
A lamp factory uses 1000 individual LEDs in a lamp. A lamp is bad if more than 50 of its LEDs are bad. What is the max allowable failure rate of the individual LEDs if the factory targets a lamp failure rate of 1e-6. (assume i.i.d. LED failure rate).
In other words, for a desired overall failure rate r, number of trials N, max number of individual failures K; I want the individual probability of failure p that satisfies binom(N,p).cdf(K) = 1-r.
I could bisect the function to find a numerical solution, but I'm hoping for a closed form solution.