Let us assume a hypothetical disease which afflicts people with 10% probability (i.e. there is a 10% chance that I have this disease right now).
Let us also assume a hypothetical test for this disease with 90% accuracy (i.e. in 9 out of 10 cases it correctly determines if somebody has the disease or not).
Finally, let us assume that I have just been tested and the result came back positive.
What is the probability that I actually have this disease?

added later: The answer is in the comments.
This kind of calculation should be interesting to doctors and their patients; but I was actually thinking about the stock market when I came up with it.
I assume investors would like to avoid market crashes and severe drawdowns, but they are quite rare: 1974, 1987, 2000, 2008 come to my mind. So they seem to happen (on average) more than ten years apart and usually the "crash phase" lasts for a few months only. In other words, the probability that a particular month will experience a "crash" is well below 10%. Therefore, if an investor wants to avoid those "disease months" consistently, she needs a predictor with accuracy much better than 90%.

added even later: Using Bayes' formula, one would calculate p(S|P) as probability to be Sick conditional on the test being Positive as p(P|S) * p(S) / p(P)
In my experience (I have asked this kind of question several times in job interviews), if people have a problem with it, the problem is usually with the denominator; i.e. p(P) = p(P|S)*p(S) + p(P|N)*p(N), with N denoting Non-sick.
So let me help the Bayesians a little bit: It is usually more intuitive to calculate the ratio p(S|P) / p(N|P) because then p(P) falls out of the equation.


I revived my homepage a few days ago.
It is minimalistic, but contains an incomplete archive of old tsm blog posts, with several broken links.
Furthermore, if you want to simulate quantum gravity (*) you can do this with the programs I published there.

The web hosting is (so far) zero cost for me and comes with only a small ad at the bottom; but in order to discourage business use, the webpage is unavailable for one hour every day, this is currently set to happen between 1am and 2am ET.

Over time I may post more stuff there - we shall see.

(*) Actually the programs calculate some statistical properties of lattice models, motivated by a search for quantum gravity. This search has not found an interesting continuum limit yet and it is unclear if and what it has to do with the correct quantum theory of gravitation ...

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