The Birthday Paradox

Last night I heard a BBC World Service programme "Unsuspected Elements" hosted by Marnie Chesterton, a friend of myself and my wife and whose father was a contemporary of mine at school and arguably the politest man in England. Marnie introduced listeners to The Birthday Paradox where a chap invites people to his birthday party. The wording of the invitation is quite detailed. It says: "Please come to my birthday party to be held on 16th February in my drawing-room at 6.00 p.m.. Those attending are not allowed to leave and all will be stuffing their faces with doughnuts injected with strawberry jam."

How many must I invite randomly to achieve a c. 50% chance of two people in the drawing-room having the same birthday each year? Hint: Marnie got the answer wrong. This error was caused by an absence of mathematical knowledge rather than manners where she has an impeccable background. She is also as good a radio broadcaster as anybody else in the BBC. As far as I know she is not betting £1,000 on Grand Albert in the 3.15 Haydock this afternoon.