An Odd Birthday Problem

As a market researcher with a particular aptitude for the quantitative aspects of the job, it’s no surprise that Alex Mullarkey today treats us to – or perhaps baffles us with! – a statistical conundrum.

When telling people that you have done a Maths degree, it provokes the most interesting reactions: some people seem impressed and interested while others look truly disgusted. I quickly learnt in my first year that if I was going to have any success with the opposite sex then I couldn’t be honest, which quickly led me to introducing myself as a dolphin trainer or astronaut (am afraid that wasn’t much better!).

However, the life of a Mathematician isn’t all bad. In fact you get to learn some truly remarkable things, such as proving that a knight (horse on the chessboard) can visit each square on the board without returning to the same square twice –see – life changing stuff! And there are some great Mathematical problems that you can wow your friends with down the pub.

The one I always find particularly interesting is the birthday problem. The problem states that in a group of 23 randomly chosen people, the probability that two people will share a birthday is more than 50%. This seems absurd as logically there are 365 days in a year (assuming it is not a leap year), so 23 people can only cover 23 of these days. Therefore, how on earth can the probability of two people sharing a birthday in a group of 23 be 50%?

As with so many of the market research projects we undertake, the way that you approach this problem is the key to understanding and solving it. If you were to take the first person on the list and check if they had the same birthday as someone else then this would allow 22 chances for a match (i.e. 22 pairings), then the second person can be paired with any of the remaining 21 people, giving an additional 21 pairings. This is then continued for the remaining people in the group, which gives a total of 253 pairs – and this is more than half of the number of days in the year. So the chance that one of these pairs has a matching birthday is quite high. If you wish to see the Maths behind the answer then the following website explains it very well:

I am sure that there will be sceptics among you; however, you certainly aren’t the first and you won’t be the last. There is a famous story where a Professor told his class about the problem. However, one student went home and couldn’t get his head around it so decided he would research the problem and prove the Professor wrong. After a few weeks of working on this problem he returned to the Professor in a huff, stating that he couldn’t work it out. The Professor decided to start off with the basics and just test it by trial and error, so they went into the first lecture theatre (conveniently of about 23 students!) and posed this problem to them. The Professor then turned towards his student and asked him when his birthday was, and the student said “24th June”, to which the Professor replied “That’s my birthday as well”.

In our office there are currently 60 workers, so using the principles of the birthday problem it is no surprise that there are two people that share the same birthday. However, it was more of a surprise that the probability there would be two people with the same birthday was 0.994 (i.e. we would have to look at 163 groups of 60 people until we had a group where no two had the same birthday). In fact, in our office there were 7 pairs with the same birthday!

Now you too can check to see if you and your colleagues share the same birthday or whether your office manages to defy the odds!

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