I've got to the point that I get really excited about interesting maths problems (this might not be a new issue, but being a teacher allows me to unashamedly embrace the inner geek), and I came across this one a couple of days ago:
A disease has broken out in the world, which infects on average one in every 1000 people. The onset has no physical symptoms in the early stages, but medics have developed a test that is able to detect the disease with an accuracy of 95%, or more precisely, the test will return positive with 100% accuracy if the person tested is infected, but will return a false positive (i.e. will say the person is infected when in fact they are not) 5% of the time if the person is not infected.
You take the test, and the result is positive. What is the probability that you are infected with the disease? Is it:
(a) 100%
(b) 95%
(c) 2%
Answers on a postcard (or a comment, if you must) with explanation. I'll reveal the answer some time in the future...
2 comments:
c) 2%
There's probably a more sophisticated way of working this out but:
If 1000 people take the test, one will have the disease. He will register positive on the test.
49.95 who do not have the disease will also register positive, because of the false positive rate
So if you take the test and the result is positive, then your probability of having the disease is 1 in 50.95, which is 1.96%, which rounds up to 2%.
That is indeed the answer - and that's probably the best way of thinking about it. Extra credit for working with accuracy, making a rounding earlier is also acceptable, considering we do have a person who is 95% infected but also 5% uninfected!
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