109 An explosion at a construction site Could have occurred

109. An explosion at a construction site Could have occurred as the result of static electricity, malfunctioning of equipment, carelessness, or sabotage. Interviews with construction engineers analyzing the risks involved led to the estimates that such an explosion would occur with probability 0.25 as a result of static electricity, 0.20 as a result of malfunctioning of equipment, 0.40 as a result of carelessness, and 0.75 as a result of sabotage. it is also felt that the prior probabilities of the four causes of the explosion at 0.20, 0.40, 0.25, and 0.15. Based on all this information, what is (a) the most likely cause of the explosion; (b) the least likely cause of the explosion?

Solution

We need to use Baye\'s theorem for conditional probability:

For two events A and B, we have:

P(A/B) = P(A/B) P(B)/ P(A)

Let A is an event of explosion and B is the specific cause.

We already know that :

B(Static electricity) ; P(A/B) = 0.25 ; P(B) = 0.2

C(Malfunction) ; P(A/C) = 0.2 ; P(C) = 0.4

D(Carelessness) ; P(A/D) = 0.4 ; P(D) = 0.25

E(Sabotage) ; P(A/E) = 0.75 . P(E) = 0.15

We need to calculate P(A) first, Using the total law of probability we get,

P(A) = P(A/B) P(B) + P(A/C) P(C) + P(A/D) P(D) + P(A/E) P(E)

Because, P(B) + P(C) + P(D) + P(E) = 1

P(A) = 0.25 * 0.2 + 0.2 * 0.4 + 0.4 * 0.25 + 0.75 * 0.15 = 0.3425

And we can calculate the probability that there was a specific cause given that an explosion occured:

B) Static Electricity : P(B/A) = 0.25 *0.2/ 0.3425 = 0.146

C) Malfunction : P(C/A) = 0.2 * 0.4 /0.3425 = 0.233

D) Carelessness : P(D/A) = 0.4 * 0.25 / 0.3425 = 0.291

E) Sabotage : P(E/A) = 0.75 *0.15 / 0.3425 = 0.33

Therefore, a) The most likely cause is Sabotage and

b) The least likely cause is Static electricity.

We further verify that P(B/A) + P(C/A) + P(D/A) + P(E/A) = 1,simply because there are only four causes for the explosion. Given that an explosion occurred, the sum of the probability for the only 4 known causes has to be 1.