Help with false positive question

• Jan 22nd 2011, 10:22 AM
Turloughmack
Help with false positive question
A blood test is 99% effective in detecting a certain disease when the
disease is present. However, the test also yields a false positive result
for 2% of healthy patients tested. Suppose that 0.5% of the population
have the disease.

(a) What proportion of healthy patients are correctly diagnosed?

(b) What is the unconditional (or marginal) probability of a positive
test result?

(c) What is the probability of having the disease given a positive test
result?
• Jan 22nd 2011, 01:24 PM
tom@ballooncalculus
Just in case a picture helps...

... with W = well, I = ill, P = testing positive, N = testing negative...

http://www.ballooncalculus.org/draw/misc/prob.png

See here Conditional probability - Wikipedia, the free encyclopedia for a more formal treatment of a very similar problem (see under the heading 'An example'), but from the tree you can see that

(b) is (398 + 99) / 20,000

and (c) is 99 / (398 + 99)

(a) is disconcertingly more obvious and (if I'm not mis-reading it) just 98%. Hope this helps.

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• Jan 22nd 2011, 01:46 PM
Turloughmack
Thanks
• Jan 22nd 2011, 10:45 PM
CaptainBlack
Quote:

Originally Posted by Turloughmack
A blood test is 99% effective in detecting a certain disease when the
disease is present. However, the test also yields a false positive result
for 2% of healthy patients tested. Suppose that 0.5% of the population
have the disease.

(a) What proportion of healthy patients are correctly diagnosed?

(b) What is the unconditional (or marginal) probability of a positive
test result?

(c) What is the probability of having the disease given a positive test
result?

I often find it helpful to suppose there are say 1,000,000 patients, 999,500 are healthy and 500 have the disease

19990 of the healthy test positive, and 495 of the diseased test positive.

So for (a) the answer is (999500-19990)/999500=0.98

For (b) (19990+495)/1000000

For (c) ...

CB