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A Tricky Probability Lesson

This lessonis not so much about how to figure the odds, but rather about how easy it is to make mistakes. Big mistakes. It starts with this simple question: If an HIV test is 98% accurate, meaning 1% of those tested will have false-positives and 1% will have false-negatives, what is the probability that you have the HIV virus if you test positive?

A. 98%

B. 99%

C. Impossible to say with this information

Probability Lesson: Watch Those Assumptions

The answer is C. It is easy to assume that the above question has the information you need to answer it. However, you can't actually say what probability of the infection is from the positive test, unless you know the underlying rate of the infection in the test group (society). You'll see why if we restate the question and answer it with an example:

Given the information above, if the underlying or "base rate" of the infection is 1%, meaning 1 in a 100 people have the infection, what is the probability you have the infection if you test positive?

A. 97%

B. 75%

C. 50%

The answer is C, 50%. Here is an example to make it clear:

If you test 10,000 people, 100 of them will have the infection (the 1% base rate). Of these, 99 will test positive and 1 will test negative (the 1% false negatives). Of the remaining 9,900 people who do not have the infection, 9801 will test negative and 99 will test positive (the 1% false positives). In total 198 people will test positive, but only 99 will actually have the infection. In other words, with a test that is called 98% accurate, half of the people who test positive don't have the infection.

The lesson is that it's easy to get confused about these things, and the accuracy rates quoted for these tests can be misleading. You have to know the base rate to make sense of them. If testing with 99% accuracy for a disease that half the people have (50% base rate), a positive test would actually mean you have a 99% probability of having the infection (4950 true-positives and 50 false positives mean that 4950 of the 5000, or 99% of the positive tests indicate the infection.

I hope I did the math right on that last example. It is easy to misunderstand these things, isn't it? That is an important lesson.

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