April 2011 question 7 -b (ii) and (iii)

An Insurance company distinguishes between 3 types of fraudulent claims:

Type 1: Legitimate claims hat are slightly exaggerated
Type:2 Legitimate claim that are strongly exaggerated
Type 3 :false claims

Every fraudulent claim is characterised as exactly one of the three types. Assume that
the probability of a newly submitted claim being a fraudulent claim of type 1 is 0.1.
For type 2 this probability is 0.02, and for type 3 it is 0.003.
The insurer uses a statistical software package to identify suspicious claims.

If a claim is fraudulent of type 1, it is identified as suspicious by the software with probability 0.5. For a type 2 claim this probability is 0.7, and for type 3 it is 0.9. Of all newly submitted claims, 20% are identified by the software as suspicious.

(ii) Calculate the probability that a claim that has been identified by the software as suspicious is:

(a) a fraudulent claim of type 1,
(b) a fraudulent claim of any type.

(iii) Calculate the probability that a claim which has NOT been identified as suspicious by the software is infact fraudulent ?

please provide solution other than mention in exam papers

 

dear sir

let event A1, A2, and A3 denotes event of Type-1 , Type2, Type -3
and P(A1)=.1, P(A2)=.2 and P(A3)=.003
And P(S/A1)= .5 , P(S/A2)= .7 ,P(S/A3)= .9,

therefore for question number 7 (ii) of a

we have P(A1/S)={ P(S/A1)*P(A1)}/p(S)
={.5*.1}/.2=.25

for part (b) I am unable to structure the data , I know that we have to use "or" between P(A1/S)or P(A2/S) or P(A3/S)

after this I am confused ???

 

Dear Sir - Part -III- not clear to me ?? please clarify

Question (iii) is some what tricky not understood?