the birthday problem, revisited

I was at work this past Thursday which was also my birthday, and I found out that my manager Mike’s birthday was also August 21st. “A good day” we both exclaimed. My company has a birthday tradition of having cake and ice cream once per month to celebrate all of that month’s birthdays, and the email had gone out to the six of us with August birthdays about what kind of cake we preferred. I was the only one who uttered German chocolate and I was swiftly outvoted by a few Key Lime Cheesecakes.

But Mike’s and my birthday landing on the same day prompted me to recall that classic birthday problem which asks: what’s the fewest number of people in a group needed so that the chance of two or more of them sharing a birthday is better than not?

So I asked my colleagues, what are the odds that any of the six of us with August birthdays share that birthday? The answer is just over 40% as it turns out. I drew up a short spreadsheet to test my answer with 10k trials. Refresh that guy a few times and you’ll see the result in S5 hover around the 40% mark.