Probability of Spotting Downed Plane in Sectors

In summary, the probability of the plane being found in either sector is (0.5*0.5) + (0.4*0.15) = 0.9.
  • #1
Spunkerspawn
9
0
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 anyone 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?
 
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  • #2


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 : )
 
  • #3


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?
 
  • #4


I think I can get closer to a solution via your suggestion. Thanks a lot!
 
  • #5


I can understand why you are struggling with this conditional probability question. It can be tricky to wrap your head around at first, but with the right approach, you can solve it easily.

First, let's define some terms that will help us understand the problem better. The probability of spotting the plane in a sector is called the "success rate" and is denoted by the letter p. In this case, the success rate for tundra is 0.5, for forest is 0.4, and for lake is 0.15. The probability of not spotting the plane is called the "failure rate" and is denoted by the letter q. In this case, the failure rate for tundra is 0.5, for forest is 0.6, and for lake is 0.85.

Now, let's tackle part (a) of the question. The probability of spotting the plane in the two tundra sectors is indeed 0.25, as you calculated. This is because we are looking for the probability of both tundra sectors being successful, which is p*tundra*p*tundra = 0.5*0.5 = 0.25.

For part (b), we need to use Bayes' Theorem, which states that P(A|B) = P(A and B)/P(B). In this case, A represents finding the plane and B represents the plane being in a certain sector. So for (i), we are looking for P(Tundra|Not Finding Plane), which is equal to P(Tundra and Not Finding Plane)/P(Not Finding Plane). The probability of not finding the plane in the tundra sectors is the failure rate for both sectors, which is q*tundra*q*tundra = 0.5*0.5 = 0.25. The probability of not finding the plane in any sector is the sum of the failure rates for all sectors, which is q*tundra + q*forest + q*lake = 0.5*0.5 + 0.6*0.4 + 0.85*0.15 = 0.575. So the probability of the plane being in the tundra sectors given that it was not found is 0.25/0.575 = 0.4348.

For (ii) and (iii), the approach is the same. The only difference is that we are
 

1. What is the "probability of spotting downed plane in sectors" and why is it important?

The "probability of spotting downed plane in sectors" refers to the likelihood of a search team or aircraft spotting a plane that has crashed or gone missing in a specific geographical area, such as a sector of the ocean or a designated search zone. This probability can be calculated using various factors, such as weather conditions, search equipment, and search patterns. It is important because it helps search and rescue teams determine the most effective and efficient strategies for locating a downed plane, potentially saving lives and minimizing search efforts.

2. How do scientists calculate the probability of spotting a downed plane in sectors?

Scientists use a variety of mathematical models and statistical methods to calculate the probability of spotting a downed plane in sectors. These methods take into account factors such as the size of the search area, the capabilities of search equipment, and the likelihood of a plane being spotted based on previous search and rescue data. Advanced technologies, such as satellite imagery and predictive modeling, may also be used to improve the accuracy of these calculations.

3. What are some factors that can affect the probability of spotting a downed plane in sectors?

There are several factors that can affect the probability of spotting a downed plane in sectors. These include weather conditions, such as fog, rain, or strong winds, which can hinder visibility and make it more difficult to spot a plane. The terrain and topography of the search area can also play a role, as mountains, forests, or other obstacles can obstruct the view of search teams. Additionally, the capabilities and resources of the search team, such as the number of aircraft and personnel, can also impact the probability of spotting a downed plane.

4. Can the probability of spotting a downed plane in sectors change over time?

Yes, the probability of spotting a downed plane in sectors can change over time. This can be due to a variety of factors, such as changing weather conditions, the progress of the search and rescue efforts, and the addition of new search equipment or resources. As more information is gathered and the search area is narrowed down, the probability of spotting a downed plane may increase. Additionally, as time passes, the chances of locating a plane may decrease due to factors such as fuel consumption, worsening weather conditions, and the possibility of the plane sinking or disintegrating.

5. How can the probability of spotting a downed plane in sectors be improved?

The probability of spotting a downed plane in sectors can be improved through various measures. These may include using advanced search and rescue technologies, such as drones, sonar, and satellite imagery, to increase the chances of spotting a plane. Improved communication and coordination among search teams can also enhance the efficiency and effectiveness of the search efforts. Additionally, incorporating data from previous search and rescue missions and ongoing research can help refine and improve the models and methods used to calculate the probability of spotting a downed plane in sectors.

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