Can you solve it? The passport control puzzle

Show me your documents please

UPDATE: The solution is now up here

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.