Coincidence and Probability

When I posted about Emila's 2010 Illustrated Calendar a few weeks ago, blogger Wan Lili of Suddenly, Heta! commented that she shares the same birthday as me. I was pleasantly surprised and noted that the chances of that to happen is quite low because this blog of mine doesn't really have that big a following.

Lili's comment reminded me of a programme I saw on The Discovery Channel some months back, about the probability of finding two persons within a group of people who share the same birthday. This problem is also known as the Birthday Paradox and there are quite a number of articles about it available on the internet. The different articles describe the problem in different ways but perhaps the simplest way that I can put it is as follows :

What is the minimum number of people needed within a group so that the odds of finding any two who have the same birthday becomes 50%?

Even for those of us who are mathematically inclined, the initial assumption we arrive at is that it must be a large number. There are 365 days in a year (ignoring leap years for simplicity)... and for 100% probability (i.e. a sure thing), we need 365+1=366 persons. Therefore, for an even chance of finding two people with the same birthday, the number is 50% of 366 or 183 persons.

This answer is wrong. Probability theory shows that we only need to gather 23 persons for the odds to become even (i.e. 50:50 chance). Now how can this be? I don't want to bore you with the mathematical analysis of this problem but you can read the links I've included below for detailed explanations.

This phenomenon is not actually a paradox in the logical sense of the word but it seems so to most people because the mathematical truth contradicts natural intuition. The human brain thinks of progression and extrapolation generally in linear terms and gets confused when some things expand on an exponential basis.

In the second reference article below, Robert Matthews and Fiona Stones carried out a study to test this theory by looking at football matches. In a football match, you can find 23 people i.e. 11 players from each team and the referee. If there are 10 such matches, then probability theory says that we should be able to find birthday pairs in 5 of them (50:50 chance). Matthews and Stones did their analysis on 10 Premier League fixtures played in 19 April 1997. It involved them checking the birthdays of 220 players and 10 referees. Sure enough, they found that there are coincident birthdays in 6 of the matches. In fact, they found two pairs in 2 of the fixtures.

To extend slightly on this subject, the theory also says that we need only 57 people for the probability of any two people with coincident birthdays to become 99%. In other words, if there is a gathering of around 60 persons, I'm willing to bet that I can find at least one pair that share the same birthday.

Graph from Wikipedia, showing the approximate probability of at least two people sharing the same birthday amongst a certain number of people

I did a bit of my own analysis on this matter using the birthdays of my Facebook friends. I compared their birthdays against the sequence of when we became friends. The result is pretty close to the theory. The 24th person who became my FB friend shares the same birthday with the 9th person. I did not have to wait long for the second pair. The 28th FB friend has the same birthday as the 22nd friend.

I was about to do the same analysis on the extended family on my wife's side (parents, siblings, in-laws, nephews, nieces etc.) but then realised that the same-birthday occurrence is found even earlier. My father-in-law (the no. 1 guy) has the same birthday as his youngest son (the 15th family member).

To use the conclusion of the Matthews & Stones research, coincidences really are “out there”, as probability theory predicts, if we take the trouble to look.

References :

1. Birthday Problem - Wikipedia
2. Coincidences : the truth is out there