Probability excercise without resolution

[Faqundo]

Hello everybody!

The next problem is picked from a guide of excercises of an University. What brought my attention is that they say that it can't been solved because there is some lack of information. It has been written in spanish so I will translate it to english. Also I will be updating my findings in the next posts.

I would like to know if you have seen similar problems or what can you say about it. Does it has a solution in your opinion?

The city X is affected by two tpes of pollution, air and water, while the Y city is only affected by air pollution.
They started a plan to control the sources of contamination.

The estimation is that the probability of controlling the air contamination successfully in the X city is 4 times greater than controlling that probability in the Y city.
Furthermore if the air pollution is controled in the Y city, the air pollution of the X city will be controlled with a 90% of probability.

The water contamination control of city X is INDEPENDENT of the air pollution control of both cities.
The probability of control both sources of contamination in X city is 0,32. Controlling the water contamination in the city X is only a half probable than controlling the air pollution in the same city (X).

Determine:

a) Probability that the air pollution can be controlled in both cities.
b) Probability that both contaminations (air and water) can be cotnrolled in both cities.
c) Probability that at least one city can be free of all sources of contamination.

These are the results proposed (they can't be wrong).
a) 0,18
b) 0,072
c) 0,448

 

[Prof]

I am not sure what you are asking for here. The answers you gave for a, b, and c are right, and there is enough information to find them. Part c is quite complicated. If you have work on parts a, b, and c, you can submit it to the forum for comments. If you need a hint to get started, first find the probability of each type of pollution in city X by using the last 3 sentences before part a. Then you can use that to find the probability of air pollution in city Y using the 3rd sentence in the problem.

 

[Faqundo]

I am not sure what you are asking for here. The answers you gave for a, b, and c are right, and there is enough information to find them. Part c is quite complicated. If you have work on parts a, b, and c, you can submit it to the forum for comments. If you need a hint to get started, first find the probability of each type of pollution in city X by using the last 3 sentences before part a. Then you can use that to find the probability of air pollution in city Y using the 3rd sentence in the problem.

Thank you Prof. I have a lot of doubts about it, I was looking for a similar problem with the resolution made to see how to outline this or any hint of how to start. Beginning from P(AX ⋂ WX) = 0,32 I tried to found one of this probabilities. But I don't have P(AX), P(WX) nor P(AX | AW) to apply the formula.

 

[Prof]

To find P(AX and WX) you know that AX and WX are independent so you can use P(AX and WX) = P(AX)P(WX). It also says that P(WX) = (1/2)P(AX).
Substitute to get 0.32 = P(AX)P(WX). Then, see if you can use P(WX) = (1/2) P(AX) in the equation with 0.32. Then you will be able to find both P(AX) and P(WX).