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The Birthday Problem amongst Friends

I continue to be fascinated by the Spooky Math of Coincidence that we have discussed in a previous blog post.

Within this particular branch of probabilities, we have the classic Birthday Problem: What is the % likelihood that two or more people share a birthday in a group of 30 people?

Well, I decided to see how things work on a large sample of people and dates - that being my own Facebook account. It occurred to me as I was writing “Happy Birthday” on people’s walls that sometimes there are two or three people with birthdays on the same day. So I decided to plough through the list and see what I could find out.

Out of 544 friends on my Facebook account, 422 have birthdays recorded (78% of all friends).

  • On 113 / 365 days (31%) no one in the sample had a birthday.
  • On 131 / 365 days (36%) just a single person had a birthday.
  • On 79 / 365 days (22%) two people shared a birthday.
  • On 36 / 365 days (10%) three people shared a birthday – that being three “pairs”, i.e. each of these people having a match with each of the other two.
  • On 5 / 365 days (1.4%) four people shared a birthday – that being six “pairs”.
  • On 1 / 365 days (0.3%) five people shared a birthday – that being 12 “pairs”.


Pie chart of birthday distribution 

So on 33.7% of the days, at least two people shared a birthday.

It seems that there are some days that are more "popular" for birthdays than others. This real-life phenomenon is not taken into account in the classic problem which assumes that all days have an equal probability.

Unfortunately my math is not quite good enough to do the fun stats, like taking different groups of 30 people amongst my friends and looking at their matches. I am looking at using freelancers on various web sites to help me out. I’m also posting my spreadsheet here so if someone fancies having a go at the data set, they are welcome!

In the meantime, I will make sure I remember to say happy birthday to someone in my group at least once every third day.