Probability - Birthday pairs

With my love of subjects around Probability and Risk, I find myself reading Simon Singh's book; Fermat's Last Theorem. I am a big fan of Simon and would certainly recommend his Book of Code to any intellectual IT persona. Coincidentally, I also bought the Book of Cod and the Book of God, both by completely different authors. Although both of these were entirely 'off subject', they threw an interesting light on the fishermen's pursuit of the Cod and mankind's pursuit of spiritual belief.

Let me provide you with an interesting story about counterintuitive Probability that I was reminded of, after reading Simon Singh's book. What is the probability of 2 people, at a party or sports gathering, sharing the same birthday. With 23 people on a football pitch or 23 people at a party you would imagine that the probability of 2 people in that group sharing the same birthday would be unlikely, given that there are 365 days to choose from. Most people would guess that there is less than 10% probability.

I will not keep you in suspense unnecessarily. The answer is just over 50%. Surprised? Well it is true. The likelihood of 2 people in a group of 23 sharing the same birthday is more than 50%. It is a counterintuitive dilemma with a counterintuitive answer.

The issue we have as humans, is that we tend to consider the problem as more complex than it is. We think that there are 23 people and 365 days in the year, so the answer must be some multiplicity of that but it is not.

The better approach is to consider the issue from a different angle entirley. The question is about pairs. How many pairs are there in such a group? To save you counting let me tell you, there are 253. The first person can be paired with 22 others, the second person can only be paired with 21 others because we have already paired up the first person. Thus, reducing the pool by 1. The third person can then be paired with 20 others and so on until we reach 253.

The odds of pairing increases dramatically when the number of people at the party increase. You will by now realise that the number of pairings with increase accordingly. The knowledge of these counterintutive issues is well known to mathematicians and your local bookmakers (bookies), one from a problematic view and the other from the view of economic gain.

So, the next time you are at a slow party, you may take the opportunity to gain kudos with the other partygoers or perhaps make some chump change. Good luck!