HIV-Testing in the desert


In a lot of popular books about statistics (at least in Germany) there is this example about HIV.

Even with a sensitivity of say 99%, the probability having HIV if the diagnosis was positiv is quite low. Indeed, it is clear since the probability P(D|p) is depending on sensitivity P(p|D), specifity P(n|nD) and prevalence P(D).

D: diseas
nD: no disease
n: negative diagnosis
p: positiv diagnosis

Now, assume the following situation.

You and another guy who is mute are in the desert and there is also a HIV-diagnosis-tool. It is only possible to do one test with this tool. You read that the tool has sensitivity 99% and specifity 95%. Just for fun your collegue does the HIV-test to himself and he gets a positiv diagnosis. He has no idea about statistics but he knows that you had a basic course in statistics. You don't know about his ***ual live and you don't know the prevalance. With other words you don't have any information about him but you know that he would very estimate if you would tell him the truth.

1a) Do you would say to him that he has HIV?
1b) If the answer is yes could you quantify this with a probability?
1c) If the answer is yes would you still have the same oppinion if the sensitivity is 90%?

Imagine that you have also a pill which would eliminate the HIV-Virus. But the pill is worth 2 million dollars which you and your collegue would get if you sell it when you left the desert.

2a) Do you would say that he has HIV?

I had to construct this story because when I asked professional statisticians they came up with Bayes (no apriori probability) and they could not give me a helpful answer. My conviction is that this individual testing without any information gives a very high probability that the tested person has HIV if the tool gives a positive test.

I hope I could make clear the problem. As you can see English is not my native language.

Thanks for help


I think I got it. If the specifity is 100 % even with a sensitivity of 10% the prediction would be 100% if the test is positiv and you need no knowledge about prevalence. The lower the specifity the more prevalence plays a role.
So, if I have no idea about prevalence and specifity, even a sensitivity of 100% could NOT tell me something about the posteriori probability of having HIV if the test is positiv.
The problem was that I didn't pay much attention to specifity.