Suppose that 1 out of 1000 people of a specific country are infected with the deadly X-virus. An inhabitant of that country goes the hospital and gives some blood for a survey. Her blood is checked on the X-virus by a test which gives a correct outcome in 99% of the cases. The test points out that the woman is infected. After the investigation, a doctor tells the woman that he is 99% sure that she is infected with the X-virus. Can the doctor be right by making this worrying statement?

If 100,000 people would be tested on the X-virus with a test which gives in 99% of the cases a right outcome, there are according to the test (100 * 0.99 + 99900 * 0.01 = 99 + 999) = 1098 people infected. If the test has a positive outcome it is right for 99 people, but wrong for 999.

Hence this implies that the probability that a person who is tested positive really is infected is only (99/1098) * 100 = 9.0%. So the doctor isn't right in making his worrying statement.