A chemical plant has an emergency alarm system. When an emergency situation exists, the alarm sounds with probability 0.95. When an emergency situation does not exist, the alarm sounds with probability 0.02. A real emergency situation is a rare event, with probability 0.004. Given that the alarm has just sounded, what is the probability that a real emergency situation exists

the probability that a real emergency situation exists is 0.16 (16%)

Step-by-step explanation:

defining the event A = the alarm sounds, we have

P (A) = probability that an emergency situation exists * probability that the alarm sounds given that an emergency situation exists + probability that a emergency situation does not exist * probability that the alarm sounds given that an emergency situation does not exist = 0.004 * 0.95 + 0.996 * 0.02 = 0.02372

then if we use the theorem of Bayes for conditional probability and define the event E = a emergency situation exists, then

P (E/A) = P (E∩A) / P (A) = 0.004 * 0.95/0.02372 = 0.16 (16%)

where

P (E∩A) = probability that an emergency situation exists and the alarm sounds

P (E/A) = probability that an emergency situation exists given that the alarm has sounded