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....

- amilapsn

- schrodinger

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- amilapsn

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

- amilapsn

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

- amilapsn

Or is it the other way round?

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## More answers

- kropot72

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

- amilapsn

I don't get it @kropot72

- amilapsn

Are you telling the given probabilities are the same?

- kropot72

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.

- kropot72

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

- amilapsn

But in the question it's 0.75

- amilapsn

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?

- kropot72

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.

- kropot72

"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.

- amilapsn

ok....

- kropot72

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

- amilapsn

Is this the answer:
A: Static Electricity
B: Malfunctioning of equipment
C: Carelessness
D: Sabotage
E: Explosion
P(E|A)=0.25

- amilapsn

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

- amilapsn

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

- amilapsn

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

- amilapsn

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

- amilapsn

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

- amilapsn

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

- amilapsn

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

- amilapsn

*should not necessarily be

- amilapsn

The given probabilities are not posterior probabilities...

- kropot72

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.

- kropot72

"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?

- amilapsn

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)

- amilapsn

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

- kropot72

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

- amilapsn

np.

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