Can you solve it? Getting through passport control

Today’s puzzle concerns passport control, the delegation of uniformed and often surly officers at ports and airports whose job it is to check your travel document is valid.

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 pretty 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 in holding cells are there because they genuinely had an invalid travel document?

[To clarify: if you are in a holding cell then either you have been found out as having an invalid passport, or you are a false positive.]

Today’s puzzle was devised by Kit Yates, senior lecturer in mathematical biology at the University of Bath and the author of The Maths of Life and Death, a book he says is about “the places in which mathematics can have a real impact on people’s lives, perhaps without them even knowing it.”

His puzzle reveals a piece of mathematics that can have a great impact on people’s lives – whether at passport control or not – and I will discuss this in the solution post that I will launch at 5pm today.

Meanwhile, NO SPOILERS PLEASE. Instead talk about your brushes with customs officials.

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 . His book The Maths of Life and Death is an accessible and thorough introduction to ideas in statistics and probability that crop up in areas such as health and justice. I recommend it.

I’m the author of Football School, a book series for 7 to 12 year olds that opens up the curriculum through football. The latest in the series, Football School Season 4, is out now. It contains chapters on the links between football and classical music, spitting, magic, timekeeping, the International Space Station, etymology and much much more.