A plane is missing and it is presumed that it was equally li

A plane is missing and it is presumed that it was equally likely to have gone down in any of three possible regions. If the plane is actually in Region 1, then a search of that region will reveal the plane with probability .75. Similarly, if the plane is actually in Region 2, then a search of that region will reveal the plane with probability .8. Finally, if the plane is actually in Region 3, then a search of that region will reveal the plane with probability .85

What is the conditional probability that the plane is in Region i (i = 1, 2, 3) given that (1) a search of Region 1 was unsuccessful and (2) a subsequent search of Region 2 was unsuccessful?

Solution

the plane is equaly likely to be in all the 3 region

hence the probability of each will be 1/3

a) since 1 region is unsuccessful it means that 1/3*0.25 = 1/12 are the chances of plane being in region 1 as only 0.75 was the chance that we can search the plane.

in region 2 it will be 1/3*4/5 = 4/15 is the chance that it will be in region 2

in region 3 it will be 1/3*0.85 = 0.280

b) if the region 2 is unsuccessful

then

probability of being in region 1 = 1/3*1/4 =1/12

probability of being in region 2 = 1/3*1/5 = 1/15 ( as we have 0.20 chance that we cannot find the plane but the plane is there)

probability of being in region 3 = 1/3*0.85 = 0.280