Question

Problem 2.128 in Section 2.10 (the missing plane): Evaluate your answers to a, b, and c if α1=0.1, α2=0.2,  and α3=0.3.

A plane is missing and is presumed to have equal probability of going down in any of three regions. If a plane is actually down in region i, let 1−αi denote the probability that the plane will be found upon a search of the i-th 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 was unsuccessful?

b: region 2, given that the search of region 1 was unsuccessful?

c: region 3, given that the search of region 1 was unsuccessful?

Answer 1

P(region 1 search was unsuccessful)

=P(plane in region 1)*P(region 1 search was unsuccessful|plane in region 1)+P(plane in region 2)*P(region 1 search was unsuccessful|plane in region 2)+P(plane in region 3)*P(region 1 search was unsuccessful|plane in region 3)

=(1/3)*(0.1)+(1/3)*1+(1/3)*1=0.7

therefore

a) P(region 1|region 1 search was unsuccessful)

=P(plane in region 1)*P(region 1 search was unsuccessful|plane in region 1)/P(region 1 search was unsuccessful)=(1/3)*(0.1)/0.7 =0.0476

b)

P(region 2|region 1 search was unsuccessful)=(1/3)*1/0.7 =0.4762

c)

P(region 3|region 1 search was unsuccessful)=(1/3)*1/0.7 =0.4762