# Matching birthdays

Mat asked me yesterday what the chances were that if 11,000 one-second-long events occurred randomly across the course of a day, two or more would overlap. Even though there are 86,400 seconds in the day to go at, the chances of none of the events overlapping are phenomenally remote.

To make calculations easier, I assumed that events occur on the second rather than starting at the millisecond level. I don’t think this simplification affects the calculations greatly, if at all.

To work things out, it’s easier to calculate the chances that all events happen in distinct slots, than it is to find out the chances that two or more events happen at the same time; take the former number from 1 and you have the latter.

Anyway, the probability that events will not collide is:

1 – ((86,400!/(86,400 – 11,000)!) * (86,400^11,000))

or generically:

1 – ((X!/(X – Y)!) * (X^Y))

where X is the number of slots and Y is the number of events.

This is so close to 1 that it doesn’t bear thinking about. I can’t calculate it, as neither my calculator nor Excel can cope with such large factorials.

The puzzle is very similar to quite a famous birthday problem. If you ask random people in the street their birthdays (day and month only), then you only need to ask 23 people before the chance of having two matching birthdays is over 50%, assuming that birthdays are evenly distributed throughout the year. The fact that there are slight deviations in birth rates throughout the year only serves to increase these odds.

If you ask 30 people, your chances go up to 71%, 40 people gives you 89%, 50 people gives you 97%, 69 people gives you a 99.9% chance of matching birthdays. At school, I shared my birthday with someone in my class of 26 (60% chance) at the age of 10, but also shared the same first and middle name. Not sure what the odds of that are!

(As an aside, the reason I can remember the number of people in the class is that I just realised that after 23 years, I can still recite the class register for that year: Allison, Aslam, Bywell, Caunt, Cheema, Chitsabesan, Crann, Cullen, Greenhalgh, Halliday, Harrison, Howarth, Jackman, Khaliq, McNeill, Mahmood, Mene, O’Neill, Pollard, Riley, Saughman (sp.), Sutcliffe, Taylor, Tilney, Troy, Verity.)