2. You do not need to worry about the possibility of the plane being elsewhere in this problem, one of the regions may be considered to include the elsewhere if there is such.
3. Let , and denote the prior probabilities that the plane is in regions 1, 2 and 3 respectivly. A look is made in region 1 and the plane not found. Let , and denote the posterior probabilities, then by Bayes theorem:
Also and which will allow the computation of and