# Conditional Probability question

#### gs1999

##### New Member
Can anyone show me how to get the correct answer from this question? I've tried drawing tree diagrams and using bayes theorem but just not sure how to compute it. Thank you so much

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#### Dason

Bayes theorem is a good idea here. Can you show what you tried.

#### Archidamus

##### Member
Almost no math needed here. It gave you the true-positive probability of the test, 91%. Its asking you what is the false-positive probability of the test. The probability of a false-positive plus a true positive must equal 1. Thus the false positive rate is 1- 0.91 = .09

#### Dason

I disagree. The .91 from the problem is P(Test positive | drug user). The question is asking for P(not a drug user | Test positive). Your reply makes it seem like you're arguing that P(A|B) = P(B|A) which is not true in general.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Bayes' rule problems are notoriously tricky - at least for me. I found it easiest to construct the classification table using a made up sample size of around 1,000 and work from there.

#### Archidamus

##### Member
I disagree. The .91 from the problem is P(Test positive | drug user). The question is asking for P(not a drug user | Test positive). Your reply makes it seem like you're arguing that P(A|B) = P(B|A) which is not true in general.
I am arguing that P(A|B) = 1 - P(A'|B) ; A being a positive testing. B being the person is a drug user.

Also, this is one of the most poorly written test questions I've ever seen. We are all interpreting it differently.

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#### Dason

The question gives us:
P(positive test | drug user) = 0.91
P(positive test | not a drug user) = 0.01
P(drug user) = 0.10

The question asks for: P(not a drug user | positive test).

I was saying that it seemed you were implying the answer is .09 because 1-0.91 = 1-P(positive test | drug user) = P(negative test | drug user).

This logic is flawed because P(negative test | drug user) doesn't equal P(not drug user | positive test) at least in general.

It turns out it gives the right answer because the way the numbers work out.

P(drug user | positive test) = P(positive test|drug user)*P(drug user)/(P(positive test|drug user)*P(drug user) + P(positive test | not drug user)*P(not drug user)) = (.91 * .10) / ((.91*.10 + .01*.9)) = 0.91.

Now you can use that 0.91 to say P(not drug user | positive test) =0.09
So we get the same answer but this time we did it properly.

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