A rare dread disease strikes one out of a million people at random.

Let's do this the manual way, so people can't argue.  (Right.)

I'll assume that 1% of the time (chosen randomly) the test results are
flipped, independent of the true result.  (Note: a test that says
nobody has the disease is 99.9999% accurate.  Which is more useful?)

Start with 100,000,000 people.

100 people have the disease.

99 people have the disease, and the test says they have the disease.

1 person has the disease, and the test says he doesn't.

99,999,900 people don't have the disease.

98,999,901 people don't have the disease, and the test says they don't.

   999,999 people don't have the disease, and the test says they do.

So there are 1,000,098 people who the test says have the disease.  Of
these, 99 actually have it, so my chance of having it is .00009899, or
roughly .01% (which would have been my first guess: I have roughly a
1% chance of false positive, which is 10,000 times as likely as having
the disease.)

Using Bayes' [Editor: "Baye's" in the original] Theorem would have been easier and faster, and people
would have argued for weeks.

-- Seth Breidbart <sethb@panix.com>

In article <335AE376.3131@su1.in.net>, "Jim St. John" <jim@su1.in.net>
writes:

|> Your doctor informs you that during a routine series of blood tests
|> you have tested positive for a very rare (1 person in a million)
|> disease.
|>
|> Hoping that the test result may be in error, you ask the doctor how
|> accurate the test is.  He replies that the test is 99.999% reliable.
|>
|> What is the probability that you have this disease?

This is a puzzle that I hope every medical student is exposed to and
understands the solution to.  A very similar puzzle once appeared in
the Journal of the American Medical Association, as part of a series
on probability in medicine.

I think the easiest (but perhaps not the best?!) way to understand the
solution is to look at how it works out for a large population.  In
this case 100,000,000 people is a convenient size.

Since only one person in a million has the disease (its "prevalence"),
there are 100 people who have it, and 99,999,900 who don't.  Of these
two groups, let's see how many test positive.  When you state that the
test is 99.999% reliable, I assume you are referring to the test's
"specificity," the probability that the test result is positive for
someone who does NOT have the disease.  Of the people who DON'T have
the disease, the number who test positive is

  99,999,900 * 0.00001  ~=  1,000.

[ Editor: specificity is the probability of a negative result for
someone who does not have the disease; the calculation uses 1 minus
the specificity, 0.00001, as the false-positive rate. ]

If we assume that the probability of a positive result for someone who
DOES have the disease (the "sensitivity") is 100%, then all 100 of the
people who DO have it will test positive.

Therefore, there are 1,100 people (on average) who test positive for
the disease, of whom only 100 actually have it.  Your chance of having
the disease is 1/11 ~= 9.1%

-- Timothy E. Vaughan <tvaughan@athena.mit.edu>, April 1997

[ Editor: from a November 1997 thread on the Monty Hall problem. ]

Michael Alan SOSS <soss@cs.mcgill.ca> wrote:
>I recall once trying to get my brother with the "1% of the
>population has disease X.  I give you a test which is 95% accurate,
>and you have disease X.  Should you be scared?" question.  I could not
>get him to understand that he only had a 16.7% (or something) chance
>of having the disease, and not 95%.

[ Editor: with 5% false positives and 5% false negatives the chance is
.95*.01 / (.95*.01 + .05*.99) = .0095/.059 = 16.1%. ]

That is because the intuitive meaning is not the correct one.  The
natural meaning of "a test which is 95% accurate" is a test which gives
the right answer 95% of the time (i.e. 5% false positives and 5% false
negatives).  If you express the probability more intuitively (i.e. "the
test is positive for 95% of people with the disease, and negative for
95% of people without it") it is harder to misunderstand.  Possibly
another reason for the confusion is that the rate of false positives
happens to be the same as the rate for false negatives.  I expect that
this is not the case for real tests (e.g. I sneeze when I have a cold,
so applying the test "Did I sneeze?" to determine if I have a cold
produces many false positives due to other irritations, but very few
false negatives).

-- Charles Bryant <ch@chch.demon.co.uk>

Yes, I once asked my father (a doctor) about this problem, and he said
that real-life tests for AIDS and the like produce far fewer false
positives than false negatives.  I.e., if you get an actual positive
result, you should be more scared than the probabilities would
otherwise indicate.

-- Robert Lacker <lacker@worldnet.att.net>