Question

An explosion at a construction site could have occurred as the result of static electricity, malfunctioning of equipment, carelessness or sabotage. Inter-views with construction engineers analysing 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 the prior probabilities of the four causes of the explosion are 0.20, 0.40, 0.25 and 0.15. Based on all the information....

Answer 1

(a) What is the most likely cause of the explosion?
(b) What is the least likely cause for the explosion?

Answer 2

Am I correct?:
"Such an explosion would occur with probability 0.25 as a result of static electricity " \(\equiv\) P(explosion/static electricity)=0.25

Answer 3

Or is it the other way round?

Answer 4

Surely "0.75 as a result of sabotage" should have been written as "0.15 as a result of sabotage". Please confirm.

Answer 5

I don't get it @kropot72

Answer 6

Are you telling the given probabilities are the same?

Answer 7

The sum of the given prior probabilities is 1. The sum of the posterior probabilities must also be 1 (by the basic rule of probabilities). The most likely error in the posterior probabilities is "...0.75 as a result of sabotage". If this is written as "...0.15 as a result of sabotage", the sum of the posterior probabilities would be 1.

Answer 8

It is quite possible for the prior and the posterior probabilities of sabotage to both be 0.15.

Answer 9

But in the question it's 0.75

Answer 10

Am I correct?:
"Such an explosion would occur with probability 0.25 as a result of static electricity " \(\equiv\) P(explosion/static electricity)=0.25
Or is it the other way round?

Answer 11

The solution is found from Bates Theorem:
\[\large P(A|B)=\frac{P(A) P(B|A)}{P(B)}\]
for proposition A, and evidence B
where P(A) is the prior, the initial degree of belief in A
P(A|B) is the posterior, the degree of belief having accounted for B
and the quotient P(B | A)/P(B) represents the support B provides for A.
So if we consider static electricity as a possible cause, the support the evidence provides for the prior is given by 0.25/0.2 = 1.25.
The support for each of the other three possible causes is found in a similar way.

Answer 12

"But in the question it's 0.75" Based on the reasons in my previous posting, you need to query this part of the question with your lecturer.

Answer 13

ok....

Answer 14

"The solution is found from Bates Theorem:" should obviously read "The solution is found from Bayes Theorem:"

Answer 15

Is this the answer:
A: Static Electricity
B: Malfunctioning of equipment
C: Carelessness
D: Sabotage
E: Explosion

P(E|A)=0.25

Answer 16

P(E|B)=0.20
P(E|C)=0.40
P(E|D)=0.15 (As you suggested)

Answer 17

P(A)=0.20
P(B)=0.40
P(C)=0.25
P(D)=0.15

Answer 18

We have to find P(A|E), P(B|E), P(C|E) and P(D|E) right?

Answer 19

P(A|E)=P(A and E)/P(E), ok?

Answer 20

But to find that we need to find P(E)...

Answer 21

Hey I think P(E|D) can be equal to 0.75...

Answer 22

Because, P(E|A)+P(E|B)+P(E|C)+P(E|D) should not be 1

Answer 23

*should not necessarily be

Answer 24

The given probabilities are not posterior probabilities...

Answer 25

The first part of the question is:
(a) What is the most likely cause of the explosion?
So you need to find what support the evidence has on each of the prior probabilities.
For static electricity the support is 1.25.
For malfunction the support is 0.5.
For carelessness the support is 1.6.
For sabotage the support is 1 (assuming the posterior is 0.15 and not 0.75).
Before taking the evidence, the leading possibilities were malfunction and carelessness, in that order. However in the light of the evidence carelessness is now the most likely cause.

(b) In the light of the evidence sabotage is the least likely cause.

Answer 26

"The given probabilities are not posterior probabilities..." The last set of probabilities are given as prior probabilities. How was it concluded that the evidence -based probabilities are not posterior probabilities?

Answer 27

Because the phrase, "...an explosion would occur with probability 0.25 as a result of static electricity..." suggests P(Explosion|Static Electricity)=0.25 \(\sf \underline{not}\) P(Static Electricity|Explosion)

Answer 28

i. e. it suggest the probability of an explosion if there is static electricity.

Answer 29

Sorry to say I need to log out now. You need to study the application of Bayes' Theorem to this kind of question. The first set of probabilities in the question are definitely posterior probabilities, based on evidence.
You could find the information here helpful:
https://en.wikipedia.org/wiki/Bayes'_theorem

Answer 30

np.