### Brain Teasers

# Drug Tests

Probability
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.Probability

10% of the people in a certain population use an illegal drug. A drug test yields the correct result 90% of the time, whether the person uses drugs or not. A random person is forced to take the drug test and the result is positive. What is the probability he uses drugs?

### Answer

The probability of event A given event B is Pr(A and B)/Pr(B).In this specific case this is

[(.1)(.9)]/[(.1)(.9)+(.9)(.1)] = 1/2.

You could think of it this way:

Say the total population is 10 people. In this case there is one drug user and 9 non-users. If the drug user gets tested 10 times, the result will be 9 positive and 1 negative. If the non-users get tested 10 times each, the results will be 9 negative and 1 positive for each user.

Therefore, if everyone is tested 10 times, there will be a total of 18 positive results: 9 for the user and 1 for each of the non-users. In other words, if you get a positive result, it is correct half of the time (9 out of 18 times in this case).

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## Comments

Jan 22, 2003

it was ok

I thought it was really good! There is a distinct lack of probability problems on this site, and this is one of the few good 'uns. Its easier to work it out like this though. Two cases : Drug user or Non-Drug user.If its drug user it will say drug user 90% of the times and if its not a drug user it will say it is a drug user 10%. Average of 10% and 90% = 50% (the same answer).

Great problem. I like the problems that you need to work on for a while.

Great teaser.

Makes sense once I read the explanation, although I was entertaining notions about an answer of 9%, i.e. 90% of 10%, even though I really don't know why, I wasn't getting anywhere with it.

Once the drug test shows positive, the percentage of known drug users doesn't have any effect on the probability that the one selected person is a drug user. It's still a 90% chance that they're a drug user at that point.

I believe the answer given is flawed, or maybe it's the way the questionsi worded. While it's true that the drug test is only valid half the time when you account for all tests done for all people, it's valid 90% of the time when you account for tests done for one single person.

I say this in case someone tries to use this on their employer when their drug test shows positive.

I believe the answer given is flawed, or maybe it's the way the questionsi worded. While it's true that the drug test is only valid half the time when you account for all tests done for all people, it's valid 90% of the time when you account for tests done for one single person.

I say this in case someone tries to use this on their employer when their drug test shows positive.

One reason you can't just average them like Smithy did is because there are far more non-drug users than there are drug users- if anything, you'd have to take a weighted average - but this still wouldn't apply because of the reason I gave in my previous post that the percentage of drug users has nothing to do with the reliability of the drug test done for one specific person.

I liked it, and it was worded fine. I've seen other problems like this that don't indicate how many false negatives there are and they are inconclusive.

If 100 people are tested, there will be 10 users(9 true positives and 1 false negative). 90 nonusers testing as 81 true negatives and 9 false positives. so being told that the the person tested positive, he is either one of the 9 true positives or one of the 9 false positives. 1/2 chance.

Good problem

If 100 people are tested, there will be 10 users(9 true positives and 1 false negative). 90 nonusers testing as 81 true negatives and 9 false positives. so being told that the the person tested positive, he is either one of the 9 true positives or one of the 9 false positives. 1/2 chance.

Good problem

Spock said "he is either one of the 9 true positives or one of the 9 false positives". While that is true, each of those outcomes (true positives or false positives) are not equally likely. There is a 90% chance that he is one of the 9 true positives. There is a 10% chance that he is one of the 9 false positives.

Consider if the problem were stated without the first sentence. It can still be answered, can't it? Therefore, the first sentence is irrelevant to the problem. I don't know how else to say it- it doesn't matter how many known drug users there are, if what you are asking is the probability for one specific person who took a drug test.

Say that person moved to another town where the percentage of known drug users was 50% instead of 10%. He takes the same drug test there. Now what's the probability? It should be the same, shouldn't it? It's the same drug test!

Consider if the problem were stated without the first sentence. It can still be answered, can't it? Therefore, the first sentence is irrelevant to the problem. I don't know how else to say it- it doesn't matter how many known drug users there are, if what you are asking is the probability for one specific person who took a drug test.

Say that person moved to another town where the percentage of known drug users was 50% instead of 10%. He takes the same drug test there. Now what's the probability? It should be the same, shouldn't it? It's the same drug test!

I may have to eat my words - I'm finally starting to see where my flaw may be.. and why probability problems are so elusive. Having more information can in reality change the probability of something.

I was erroneously stuck on trusting the drug test to override any other information. It actually DOES matter how many known drug users there are, however non-logical that seemed to my funky brain..

If you ignore the drug test results, the chance they use drugs is 10% according to the population. If you ignore the population, the chance they use drugs is 90% according to the drug test. But when you account for both, the chance is 50% as you all have explained. Each of the outcomes outlined are equally likely after all, if you account for the probability from the population as well as the drug test.

I guess I was trying to make it a more real world scenario, in which you don't know for sure the percentage of known drug users, so you have to just rely on the test. Given that in this fictional scenario we somehow know for certain the percentage of known drug users in the population, that information does in fact change the overall probability.

But if the way that you "know" that 10% of the population use drugs is by using that drug test, than it's not a definite statistic, so that would change everything. Then I think the answer would go back to 90%.

I was erroneously stuck on trusting the drug test to override any other information. It actually DOES matter how many known drug users there are, however non-logical that seemed to my funky brain..

If you ignore the drug test results, the chance they use drugs is 10% according to the population. If you ignore the population, the chance they use drugs is 90% according to the drug test. But when you account for both, the chance is 50% as you all have explained. Each of the outcomes outlined are equally likely after all, if you account for the probability from the population as well as the drug test.

I guess I was trying to make it a more real world scenario, in which you don't know for sure the percentage of known drug users, so you have to just rely on the test. Given that in this fictional scenario we somehow know for certain the percentage of known drug users in the population, that information does in fact change the overall probability.

But if the way that you "know" that 10% of the population use drugs is by using that drug test, than it's not a definite statistic, so that would change everything. Then I think the answer would go back to 90%.

This is slightly wrong. The Probability is 52% please check Bayes Theorem in wikipedia:

http://en.wikipedia.org/wiki/Bayes%27_theorem #Example_1:_Drug_testing

http://en.wikipedia.org/wiki/Bayes%27_theorem #Example_1:_Drug_testing

This is actually an important one. Look up "base rate". It is relevant to medical tests. If a condition is rare, then a test can only find a few people (true positives) with it, whereas the number of false positives will be more as a few percent of a huge number is still a lot. This is a problem with screening programmes sometimes; if everyone who tests positive then has some further investigation you will be doing expensive and possibly dangerous things to people who don't need it. But if a condition is common, then the number of true positives may be much larger than the number of false positives. It depends on the base rate, which here is the actual rate of drug use.

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