# Conditional probability - Probability of spotting a downed airplane (really basic)

## Main Question or Discussion Point

Hey guys, first of all I want to say hey to all! I have been a long time lurker and follower of these forums and have been known to find a lot of your answers helping me out throughout my college life.

I also know that you just don't like to spit out the answers for those who don't put in the effort so here it goes:

This is a very basic conditional probability question, but I can't seem to find a trick in solving these questions and they always get me. Any help would be greatly appreciated. The question is as follows:

The air rescue service of the armed forces divided the flight path of a downed plane into search sectors. In one mission a search plane will overfly every square kilometer of the designated sector. From past experience, one knows however that in one mission there is only a 50% chance of spotting a plane down in a sector of tundra. There is a 40% chance of spotting a plane down in a forested sector and a 15% chance of spotting a plane down in a lake sector.
(a) A small plane is down in an area containing two sectors of tundra, one lake sector and one forested sector and a priori the plane has an equal chance of being down in any one of the fours sectors. The mission director decides to use the first mission to search the two sectors of tundra. What is the probability of spotting the downed plane?

(b) Given that we did not find the plane in the tundra sections, what is the probability that the plane is
(i) in the tundra sections?
(ii) in the forested section?
(iii) in the lake sector?

Ok, so for a) what I did was basically the plane is going over the two sectors of Tundra which has a probability of 50% of containing the downed plane. So P(T) . P(T) would be 0.5*0.5 = 0.25 is the probability of finding the plane in the two sectors of Tundra, right?

for b)
(i) I said P(Finding Plane | Tundra) = P(Finding Plane intersect Tundra) / P(Finding Plane)
(ii) P(Finding Plane | Forest) = P(Finding Plane intersect Forest) / P(Finding Plane)
(iii) P(Finding Plane | Lake) = P(Finding Plane intersect Lake) / P(Finding Plane)

P(Finding Plane) would be the total probability of finding the plane in each sector so given that it has an equal chance of being in either sector 2/4 * P(Tundra) + 1/4 * P(Forest) + 1/4*P(Lake)

I am not sure about my answers nor my approach to this problem. Any tricks or tips to solve this problem or conditional probability in general?

Related Set Theory, Logic, Probability, Statistics News on Phys.org

I think I should have posted this to the homework section, it reads much more like a textbook style question. So if one of the mods can move this question, or if anyone would still like to help out that would be appreciated : )

chiro

Hey Spunkerspawn and welcome to the forums.

I'll start the ball rolling by considering part a) first.

For this question we have to know two things: the first thing is the probability that the plane will actually be in a particular area, and the probability that the plane will be found in a particular area.

You are making the assumption that the probability of the plane existing in a sector (which is not the same as the probability of the plane being found: remember they are two completely different things although possibly related in some way) is 0.5 for each sector.

However you haven't incorporated the probability of finding the plane in your particular sectors. This is the conditional part and where it comes in.

So given that you have P(Found|Sector1 and Sector2) to be calculated, you have to use the information of P(Found|Sector1) and P(Found|Sector2) where Sector1 and Sector2 are independent areas and are to be treated independently as probability measures.

Based on this information, the information given to you in the question, and Bayes Rule can you either solve a) or get closer to a solution?

I think I can get closer to a solution via your suggestion. Thanks a lot!