Question

5. (10 pts.) Suppose, for example, that an explosion at a chemical plant could have occurred as a result of any of three mutually exclusive causes: equipment malfunction carelessness, or sabotage. It is estimated that such an explosion could occur with a probability of 0.2 as a result of equipment malfunction, 0.4 as a result of carelessness, and 0.75 as a result of sabotage. It is also estimate that the prior probabilities of the three possible causes of the explosion are, respectively, 0.50, 0.35, and 0.15. Use Bayes' theorem to determine the most likely cause of the explosion

Answer 1

Ans:

Given that

P(explosion/malfunction)=0.2

P(explosion/carelessness)=0.4

P(explosion/sabotage)=0.75

P(malfunction)=0.5

P(carelessness)=0.35

P(sabotage)=0.15

Now,

P(explosion)=0.2*0.50+0.4*0.35+0.75*0.15

=0.3525

Use Bayes theorm:

P(malfunction/explosion)=P(explosion/malfunction)*P(malfunction)/P(explosion)

=0.2*0.50/0.3525=0.2837

P(carelessness/explosion)=0.4*0.35/0.3525=0.3972

P(sabotage/explosion)=0.75*0.15/0.3525=0.3191

As,P(carelessness/explosion) is highest,so most likely cause is carelessness.