When the Birthday Coincidence Goes Wrong
A lesson in Risk Management
29 November 2013
For a few years, one of my favourite maths demonstrations for big audiences has been playing a game of Birthday Bingo.  I select a segment of the audience - at least 50 people - and then announce that I can somehow sense that two of those audience members share a birthday.  I then go further by telling them that I will ask each of them to announce their birthday in turn, and that by the time I have reached the seventh or eighth person, I predict there will already have been a shout of Bingo! from somebody in the cluster of 50.
It has never failed, a matching pair always crops up, usually by the time I've reached the fifth person. And everyone is suitably amazed.  Until yesterday that is.  And unfortunately, it happened in front of a seriously unimpressed audience of 600 teenagers at a theatre in Birmingham.  As my group, I picked the front row of the Dress Circle, which in the morning had contained 53 people.  Unfortunately, the afternoon show was not quite as packed, there were gaps in the front row, and it turns out there were only 40 students.  With 50 people the chance of there being at least two shared birthdays turns out to be around 97% (so it will only fail 1 time in every 30).  You can work this out by calculating the chance that all 50 people have DIFFERENT birthdays, which is 1 x 364/365 x 363/365 x...etc. all the way to 316/365.  Multiply those numbers out and you get about 0.03 (=3%), and if the chance of no coincidence is 3% then the chance of at least one coincidence is 97% .  In fact the odds are even better than that because birthdays aren't evenly distributed across the year.  For example there are more birthdays in September (which gives a hint as to what goes on at Christmas parties).
Reduce the size of the group to only 40, however, and the odds plunge massively.  OK, so the chance of success is still around 90%, but that means one time in ten it will go wrong.  And yesterday it did.  I was already feeling concerned by the time I got to the tenth person, and with good reason.  I've since worked out that if the first 10 selected from a group of 40 don't have a birthday shared by somebody else in the group, the chance that the experiment will fail has trebled, to around 30%.  And when it does go wrong, checking every single birthday for the rest of the group becomes a rather desperate exercise.
It has been an important lesson in risk management.  Next time I will double-check that my quota is above 50.  And the bigger the audience, the more contingency I will need (a failed experiment can be laughed off in a small room, but looks crass in a big theatre).  Just increasing the group from 50 to 60 reduces the chance of failure from 1 in 30 to about 1 in 200.  And that is a risk I can cope with.