Calculating Probability of Naturally Polluted Lake

[cue928]

If I know the probability of a lake being polluted is .14 and the probability of it being naturally polluted is .25, how do I find the probability that it is naturally polluted? I have played with this for a long time and still have no idea how to calculate it. In fact, I'm not even sure where to start. Any guidance would be greatly appreciated.

 

[nonequilibrium]

the probability of it being naturally polluted is .25, how do I find the probability that it is naturally polluted

Huh? You already stated what it is

 

[cue928]

That's what I thought but it can't be that simple. It's in the section on probability rules for unions and intersections.

 

[Stephen Tashi]

If I know the probability of a lake being polluted is .14 and the probability of it being naturally polluted is .25, how do I find the probability that it is naturally polluted?

My guess is that you misstated the problem. You gave 0.25 as the probability that the lake is naturally polluted and then you asked what that probability was.

You probably meant to say "The probability of a lake being polluted is 0.14. Given that a lake is polluted, the probability that it was naturally polluted is 0.25. Find the probability that a lake is naturally polluted."

$A =$ the set of cases where the lake is polluted
$B =$ the set of cases where the lake is naturally polluted

$P(A \cap B) = P(A|B)P(B) = P(B|A) P(A)$

Our interpretation of language tells us that $B \subset A$
Hence in this problem $A \cap B = B$

 

[cue928]

So in our boo it shows the multiplicative rule for independent events as P(A intersect B) = P(A) * P(B), so is it .24 *.14 = .035?

 

[cue928]

Another clarifying question: if you have a probability of success of .95 and two different actors acting independently:
1) Probability that both are successful? I assumed this was P(A intersect B) but did not know how to solve it?
2) Probability that neither are successful? I did 1-.9025 = .0975
3) Probability that either are successful? I assumed this was P(A U B) = .95*.95 = .9025

 

[Stephen Tashi]

No. A and B are not independent events. B can't happen unless A does. You better read what your book says about "conditional probability".

 

[Stephen Tashi]

You are confusing situations where a problem speaks of "A and B" with situations where the problem speaks of "A given B". The probability of "A and B" is the same as the probability of "A intersect B". If A and B are independent, you may multiply P(A)P(B) to get the answer.

In the problem of the lakes, your arithmetic is mostly correct (you have a typo of ".24" instead of ".25" ) , but you are using the wrong reasoning to justify it. You haven't stated the original problem precisely. If you don't pay enough attention to the wording of a problem to state it correctly, then you will have a hard time interpreting its details. There are many clever ways that problem can talk about "A given B" without actually using the word "given".