The probability that there could be a fire an apartment buil

The probability that there could be a fire an apartment building is 0.004. In case of fire, the alarm correctly detects it with probability 0.975. The probability of false alarm (there is alarm but no fire) is 0.015. Suppose, the alarm goes off, what is the probability that there is fire in the building?

Solution

Let

A = alarm rings
F = there is fire

Hence,

P(A) = P(F) P(A|F) + P(F\') P(A|F\') = 0.004*0.975 + (1-0.004)*0.015 = 0.01884

Hence,

P(F|A) = P(F) P(A|F) / P(A) = 0.004*0.975/0.01884 = 0.207006369 [ANSWER]