A plane is missing and is presumed to have equal probabilityof going down in any

Question

A plane is missing and is presumed to have equal probabilityof going down in any of three regions. If a plane is actually downin region i, let 1-i denote the probability thatthe plane will be found upon search of the ith region, i=1,2,3.What is the conditional probability that the plane is in a) region 1, given that the search of region 1 wasunsuccessful b) region 2, given that the search of region 1 wasunsuccessful c) region 3, given that the search of region 2 wasunsuccessful A plane is missing and is presumed to have equal probabilityof going down in any of three regions. If a plane is actually downin region i, let 1-i denote the probability thatthe plane will be found upon search of the ith region, i=1,2,3.What is the conditional probability that the plane is in a) region 1, given that the search of region 1 wasunsuccessful b) region 2, given that the search of region 1 wasunsuccessful c) region 3, given that the search of region 2 wasunsuccessful

Explanation / Answer

We are given P(Fi/Ri)=1-i      hence P(F'i/Ri)=i  P(Ri)=1/3 a)P(R1/F1')=P(F'1/R1)P(R1)/P(F1')=1(1/3)/P(F'1) Now P(F1')=P(F1'andR1)+P(F'1 and R2)+P(F'1andR3)=P(F'1/R1)P(R1)+P(F'1/R2)P(R2)+P(F'1/R3)P(R3) =(1/3)+1/3+1/3=(1+2)/3 Therefore P(R1/F'1)=1/(1+2) b)P(R2/F1')=P(F'1/R2)P(R2)/P(F'1)=1(1/3)/[(1+2)/3]=1/(1+2)
c)P(R3/F2')=P(F'2/R3)P(R3)/P(F'2)=(1)(1/3)/[(2+2)]=1/(2+2)

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