The military operates an early-warning system to detect incoming missiles. It consists of
n detectors, which are inspected once per month and repaired at that time if necessary. Between
inspections, there is a 5% chance that a detector will malfunction. If the system can only detect
incoming missiles if at least four detectors are working properly, how many detectors should the military
install if they want to be 99.9% certain that the system won't break down between inspections?

Question

The military operates an early-warning system to detect incoming missiles. It consists of
n detectors, which are inspected once per month and repaired at that time if necessary. Between
inspections, there is a 5% chance that a detector will malfunction. If the system can only detect
incoming missiles if at least four detectors are working properly, how many detectors should the military
install if they want to be 99.9% certain that the system won't break down between inspections?