A screening programme tests ten thousand people for a condition that one person in a hundred actually has. The test is good. It detects ninety-nine percent of the people who have the condition, and it wrongly returns a positive result for only five percent of the people who do not.
Count the results. A hundred people have the condition, and ninety-nine of them test positive. Nine thousand nine hundred people do not have it, and five percent of them — four hundred and ninety-five people — test positive anyway.
So five hundred and ninety-four people receive a positive result, and only ninety-nine of them are ill. If you are one of those five hundred and ninety-four, your chance of having the condition is ninety-nine out of five hundred and ninety-four — about one in six.
Nothing here is a trick, and the test is not bad. The number that surprises people comes from the denominator. "Ninety-nine percent accurate" describes the people who have the condition. Whether you have it, given a positive result, is a completely different question, and its answer depends on how rare the condition is.
The programme is still worth running. Before it, ten thousand people were unexamined; after it, the search has a manageable shape.
This is why a positive screening result is normally followed by a second, more specific test rather than by treatment. The first test narrows ten thousand people down to six hundred. The second one is work that only has to be done on six hundred.