Risk and probabilities can confuse our reasoning.

 

You are given the following information:

(a) In random testing, you test positive for a disease.

(b) In 5% of cases, this test shows positive even when the subject does not have the disease.

(c) In the whole population, one person in one thousand has the disease.

 

What is the probability that you have the disease?

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Many people reply: 95%  (even scientific experts).   This is not quite right. The answer is about 2%.

 

To see why, the information in (c) is crucial.  Consider a population of 1,000 people.  Of these, on average, one will have the disease.  

 

But, 5% = fifty others will test positive, without having the disease.   Of those who test positive, therefore, only one really has the disease and 50 have tested positive falsely.    So out of 51 people testing positive, 1 person, which is almost 2%; will turn out to have the disease. 

 

( 2% is much better than 95% for diseases.  You wouldn't want to get it wrong.  It could ruin your whole day. )