### Brain Teasers

# Drug Tests

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

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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