Earlier today I set you the following question:
Let’s say that 1 in 10,000 people who present themselves at UK passport control have invalid passports, and let’s say that UK passport control is very good at detecting invalid passports. When presented with an invalid passport, an officer will pick this up 99 times out of 100. Travellers found with invalid documents are sent to a holding cell. Let’s also imagine that UK passport control will, out of caution, very occasionally send someone to the holding cell whose passport is perfectly valid. These “false positives” occur just 0.1 per cent of the time.
What proportion of travellers sent to holding cells are there because they genuinely had an invalid travel document?
Solution About 9 per cent.
In other words, if you arrive in the holding cell, it is much more likely that you are there by mistake.
Similarly – in certain types of screening for serious diseases, if your test comes up positive then it is much more likely that you don’t have the disease. I’ll explain why after I do the maths:
The easiest way to solve the problem above is to imagine 1,000,000 hypothetical travellers. Of these 1 in 10,000 – that’s 100 people – will be travelling without a valid passport. 99 of these 100 people will be picked up by the authorities (and one will slip the net). The remaining 990,000 travellers will have valid documentation. Of these 0.1% - or 990 – will end up in the holding cells. In total there will be 1089 people in the holding cells and 990 of them will be there for the wrong reason. Only 99/1089, or roughly 9%, will have incorrect documentation.
This surprisingly low proportion of hypothetical correct detections has important real-world consequences for screening programmes. When the prevalence of a disease, like breast cancer, is low in a population (an undiagnosed breat cancer rate as low as 0.4% in women over 50 who regularly attend screening) then even a relatively low rate of false positives (roughly 10% for breast cancer screening) will mean that the vast majority of women recalled for further testing will not have the disease.
No test is perfect – and false positives are often a lot more common than true ones.
I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
Thanks to Kit Yates for suggesting today’s puzzle . If you want to know more about important ideas in maths that crop up in areas like health and justice, go out and but his impressive debut, The Maths of Life and Death.
