January 16, 2009

Probability Puzzle

If a test to detect a disease whose prevalence is one in a
thousand has a false positive rate of 5%, what is the chance
that a person found to have a positive result actually has
the disease, assuming you know nothing about the person’s
symptoms or signs?

Secret Squïrrel said...

A false +ve rate of 5% means that for every 100 people without the disease who are tested, 5 of them will be incorrectly reported as having the disease.

If you test one million people, 50,000 will incorrectly be reported as having the disease. Since the disease occurs in 1 in 1,000 people, 1,000 of the million will have the disease. The question says nothing about the false NEGATIVE rate of the test so I will assume that it is insignificant.

So, out of one million people tested, 51,000 will show as having the disease (1,000 infected + 50,000 false +ves).

Therefore the chance of actually having the disease if you receive a +ve result is 1,000/51,000 or 1.96%

Anonymous said...

WOW! 1.96%??? LOL! I just love it when someone overcomplicates the answer!

The prevalence of the disease is completely irrelevant! We dont care about the nagatives. 5% of the positives are false, thus 95% of the positives actually have the disease. Nice try!

Secret Squïrrel said...

Evan, you clearly do not understand what "false positive" means. If you did you would know that the prevalence of the disease is relevant.

And you need to learn some manners.

Anonymous said...

I have to agree with anonymous; 5% of POSITIVE results are false. Sorry, Squirrel you botched this one.

Anonymous said...

Is 'Evan' the Joker and you're Batman?

Agent Mayhem said...

Squirrel is most certainly correct. A false positive means that someone who DOESN'T have the disease shows up as actually having it.

Imagine a million people that we KNOW do NOT have the disease. How many of them will show up as positive if they take the test?

Now imagine a perfect test where there are never any false results. One million people take that one. How many show up positive?

It should be obvious that the first group would far outnumber the second.

Anonymous said...

Squirrel is incorrect. Read A Drunkard's Walk, a book about random numbers. The author was tested false positive for AIDS. A false positive means that someone who DOESN'T have the disease has a test that says he has the disease, but DOES NOT.

E

Anonymous said...

Secret Squirrel is right. The question was presented by Cassels and Al. to 60 Harvard Medical School staff and students in 1978. It was used as an example to show the conditions where base-rate data are neglected by Daniel Kanhneman (Nobel prize winner) and the late Amos Tversky. The answer is 2% as indicated page 154 of Judgment Under Uncertainty: Heuristics and biases. Edited by Kahneman, Slovic and Tversky.

Jason said...

Many of you got pwned by Squirrel.

Keaysovxy said...

I have to agree with anonymous; 5% of POSITIVE results are false. Sorry, Squirrel you botched this one.

Anonymous said...

Secret Squirrel can't be right. If the test were 100% accurate (no false positives) his/her calculations would result in just a 2% chance that a person with a positive result would have the disease.

But that contradicts the fact that the test is 100% accurate. A positive result means you must have the disease.

It would also make tests like this completely useless in the real world. You could safely ignore all test results for rare diseases, regardless of how accurate they were.