Two detectors both fire on a rare event

Question

A certain event occurs on 5\% of monitored intervals. Two independent detectors each fire on 80\% of intervals when the event is truly present, and each falsely fires on 10\% of intervals when it is absent. On one interval, both detectors fire. What is the probability the event is truly present?

Accepted Answer

Imagine 1{,}000 intervals: 50 with the event, 950 without. The detectors are independent given the truth, so multiply their rates. - Event present, both fire: 50 	imes 0.80 	imes 0.80 = 32. - Event absent, both fire: 950 	imes 0.10 	imes 0.10 = 9.5. Of the 32 + 9.5 = 41.5 double-fires, 32 are real: P(	ext{present} \mid 	ext{both fire}) = \frac{32}{41.5} \approx 0.771 = 77\%. A single fire gives only about 30\%, but two independent fires make a false alarm quite unlikely (it needs both detectors to err at once), so belief jumps to about 77\%. Takeaway. Requiring two independent alarms sq…

Two detectors both fire on a rare event

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