The Birthday Paradox


While procrastinating this morning (I was putting off studying New Testament Greek), I started thinking about people with the same birthday as me. I realised that there are quite a few: at least two of my relatives share my birthday, a neighbour when I was a child, my manager from my last job, and one of my fellow seminarians here in Rome. I even know someone (a very good friend, in fact!) who was not only born on the same day but also in the same year.

This led me to do research on “same birthdays” online and I came across a probability theory called the birthday paradox. According to this phenomenon, if there are 20 people in a room, there’s a 50/50 chance that the two of them will have the same birthday. In a room of 75, there’s a 99.9 chance of people matching. I was surprised by this: there are 365 days in a year (ignoring leap years) so surely the probability of finding someone with the same birthday is much lower?

A confession: I hated Mathematics at school (still do!) and it was the reason why I focused on English instead. Hence, I am not the right person to explain the birthday paradox and so I turn to the website howstuffworks which explains it very well:

The reason this [the birthday paradox] is so surprising is because we are used to comparing our particular birthdays with others. For example, if you meet someone randomly and ask him what his birthday is, the chance of the two of you having the same birthday is only 1/365 (0.27%). In other words, the probability of any two individuals having the same birthday is extremely low. Even if you ask 20 people, the probability is still low — less than 5%. So we feel like it is very rare to meet anyone with the same birthday as our own.

When you put 20 people in a room, however, the thing that changes is the fact that each of the 20 people is now asking each of the other 19 people about their birthdays. Each individual person only has a small (less than 5%) chance of success, but each person is trying it 19 times. That increases the probability dramatically.

So the next time you’re in a room of 20 people, test it!