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Clarification on Bayes' Theorem
ASK MARYLIN: FALSE POSITIVES
In one of Marilyn Savant's columns in parade Magazine the following question was asked.
Suppose we assume that 5% of the people are drug-users. The test is 95% accurate, which we'll say means that if a person is a user, the result is positive 95% of the time; and if s/he isn't, it's negative 95% of the time. A randomly chosen person tests positive. Is the individual highly likely to be a drug-user?
First suppose that there are 10000 people in the population.
"5% of the people are drug-users" implies:
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Drug User |
Not Drug User |
||
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Test Positive |
|||
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Test Negative |
|||
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500 |
9500 |
10000 |
" If a person is a user, the result is positive 95% of the time" implies:
|
Drug User |
Not Drug User |
||
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Test Positive |
475 |
||
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Test Negative |
25 |
||
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500 |
9500 |
10000 |
"If s/he isn't, it's negative 95% of the time" implies:
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Drug User |
Not Drug User |
||
|
Test Positive |
475 |
475 |
|
|
Test Negative |
25 |
9025 |
|
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500 |
9500 |
10000 |
Therefore:
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Drug User |
Not Drug User |
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Test Positive |
475 |
475 |
950 |
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Test Negative |
25 |
9025 |
9050 |
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500 |
9500 |
10000 |
A randomly chosen person tests positive. What is the probability that s/he is a drug-user?
A randomly chosen person tests negative. What is the probability that s/he is not a drug-user?
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