Three Rock Mountain

Saint Anthony Maria Zaccaria

I received an interesting e-mail from a bloke on MySpace recently. Entitled 'Reclaim Your Birthday', it attached a spreadsheet which contained a list of days of the year against which previous recipients of the spreadsheet had marked in their birthdays. Now obviously broadcast by e-mail obviously mitigates against the maintenance of a single coherent list, as the list participants would diverge at each node in the thread, but that's not the point.

While we take special notice of our own birthdays, nobody else is really bothered, family and close friends excepted, because it doesn't affect them or the world in which they live (unless you're a reigning monarch). And yet it's pleasing when others remember or mark them. A random act of consideration as it were, it means nothing but a buoy to your self-esteem.

Surprisingly this is something that MySpace has latched onto. It sells itself as a social networking site, and encourages participants to mutually recognise each other's sites as 'friends'. I have many more friends on MySpace than I do in real life, the difference being that a friend in real life has accepted me for who I am rather than who I appear to be. No, actually that's probably not true. Friends in real life are people you relate to on an ongoing basis. No that's not true either. Some of my dearest friends I may not see from one end of the year to another and some of the bloggers I've met through my internet excursions this year interact with me daily, even though I wouldn't recognise them if I passed them in the street.

OK, sod it. My definition of a friend is somebody who would lend me a tenner if asked.

Anyways, whatever the qualification, it's nice to see MySpace promoting friendship as a concept, even if in rather a shallow fashion. A nice development from this is that if you choose to enter your birthday as part of your profile details, it will charmingly send around a reminder to all of your 'friends' that you have a looming birthday a couple of days in advance of the big event. If they're interested, they may take the time to drop you a note of congratulations, which I have to admit is an act of genuine friendship if not memory power.

This takes me back to this notion of reclaiming your birthday. Growing up Catholic in Ireland when it was still a theocratic nation, one of the things you learned real fast was that your birthday didn't belong to you, it belonged to some saint who owned the day and obligingly let you be born on it. In my case this was Saint Anthony Maria Zaccaria. I'm even named after him in part. Irish Catholicism: You're only here by the grace and forbearance of others and even then you carry Original Sin. Talk about making you feel unwelcome!

Anyways, I didn't fill in my birthday on the spreadsheet, because I already know when it is. Besides, my birth date had already been claimed by somebody other than Saint Anthony Maria Zaccaria and I wasn't about to fight them for it. But the mail reminded me of a statistical quirk I studied back in college many years ago.

For any given body of people, what are the odds that at least one pair share a common birthday? Now obviously, hard logic dictates that you need 366.25 people to be completely certain that you have such a pairing, but in the murky world of statistics, the gap between zero and certainty fills up quite quickly. Consider the following progression as people file into an imaginary party round at Three Rock Mountain Manor:
  • 1 person in the room: Zero percent chance of a match, because there's only the one punter. Rigid logicians will contend that this constitutes a valid matching set, because by definition this person shares their own birthday with themselves. However rigid logicians aren't invited to this type of party, so let's ignore them.
  • 2 people in the room: They don't really know each other, so small talk turns to the subject of birthdays. Leaving leap years to the rigid logicians to debate among themselves (and remembering that that's the kind of behaviour that gets them uninvited from these parties), the chances of these two (let's call them A & B) having a common birthday is 1/365 or 0.274%.
  • 3 people in the room: Enter C. The odds of A and B having a common birthday is 1/365 or 0.274%. Add to this the odds of A and C having a common birthday (again 1/365) and the odds of B and C having a common birthday (1/364 since we know A and B don't have a common birthday). The odds of a single common pair when there are 3 people in the room has risen to (2/365)+(1/364) or 0.82%.
  • 4 people in the room: Enter D and the number of possible pairings starts to rise exponentially: A+B, A+C, A+D all have a 1/365 shot. Following on from this B+C and B+D both have a 1/364 shot since none of them share A's birthday. Lastly C+D have a 1/363 since they don't share common birthdays with A or B. The likelihood has risen to (3/365)+(2/364)+(1/363) or 1.65%, double the odds when there was only one less person in the room.
  • 5 people in the room: You can guess the pattern for here once E arrives on the scene. P= (n-1/365)+(n-2/365-1)+(n-3/365-2)++(n-(n-1))/(365/n) for all iterations n=1 up to N. 2.75% probability to be precise
By the time 10 people have arrived at the party the probability of at least one common birthday has risen to 12.4%. If there are 14 people in the room, the probability has risen to over 25%. The probability exceeds 33% with only two more attendees and exceeds 50% with as few as 20 people in the room. 24 attendees sees the probability exceed 75% and then it starts to get silly. Still, probably worth betting a tenner on next time you're at a party and conversation starts to falter.

© Kevin O'Doherty 2007