bayes theorem

Our first screening example featured a fictitious terrorist-detecting device that was not very accurate. However, as this real example shows, even when the tests are incredibly accurate we still get some surprising results if the underlying condition is rare.

A secret government agency has developed a scanner which determines whether a person is a terrorist. The scanner is fairly reliable; 95% of all scanned terrorists are identified as terrorists, and 95% of all upstanding citizens are identified as such. An informant tells the agency that exactly one passenger of 100 aboard an aeroplane in which in you are seated is a terrorist. The agency decide to scan each passenger, and the shifty looking man sitting next to you tests positive. Were you sitting next to a terrorist? What are the chances that this man really is a terrorist?

Since 1988, women over 50 in the UK have routinely been offered screening for breast cancer, even if they have no other symptoms, and in 2004/05 1.7 million women were screened. Those with a positive mammogram are recalled for further investigations, at considerable cost in anxiety, resources and pain and discomfort. But how many of these women really have breast cancer?