The Birthday Paradox

What are the odds?

July 27, 2021

The next time you are in a group of 30 or more people and you want to have an icebreaker activity, have the group line up by the day of the month they are born. Odds are 70 percent that two of the people in the group will share the same day.

This is a phenomenon known as the Birthday Paradox or Birthday Problem. It’s all based on exponents and probabilities. According to the Infallible Wikipedia (who got the information from a whole bunch of smart scientists), this is how it works:

“In probability theory, the birthday problem or birthday paradox concerns the probability that, in a set of n randomly chosen people, some pair of them will have the same birthday. In a group of 23 people, the probability of a shared birthday exceeds 50%, while a group of 70 has a 99.9% chance of a shared birthday. (By the pigeonhole principle, the probability reaches 100% when the number of people reaches 367, since there are only 366 possible birthdays, including February 29.)

These conclusions are based on the assumption that each day of the year is equally probable for a birthday. Actual birth records show that different numbers of people are born on different days. In this case, it can be shown that the number of people required to reach the 50% threshold is 23 or fewer.

The birthday problem is a veridical paradox: a proposition that at first appears counterintuitive, but is in fact true. While it may seem surprising that only 23 individuals are required to reach a 50% probability of a shared birthday, this result is made more intuitive by considering that the comparisons of birthdays will be made between every possible pair of individuals. With 23 individuals, there are (23 × 22) / 2 = 253 pairs to consider, which is well over half the number of days in a year (182.5 or 183). (snip)

The history of the problem is obscure. The result has been attributed to Harold Davenport; however, a version of what is considered today to be the birthday problem was proposed earlier by Richard von Mises.”

Personally, my brain kinda goes ‘tilt’ when I see cryptic scientific characters and formulas which show me how to calculate all of this. So I leave that to you brainiac statistics folks and share my own personal experience with this phenomenon.

The first time I encountered this was as a 13 year old in my 8th grade English class. More about that in a bit. Often, when I’m in a group situation and looking for a way to engage people in conversation, I will ask them their birthday (not the year, just the day) and talk about the paradox. This will often get others interested and soon the entire group is comparing days until, and it usually happens, we find the pair with the same birthday.

Over the years I have been the person who matches another who shares my birthday fairly often. I can think of at least five times this has occurred.

But it was that first time which I think might have the odds makers scrambling to figure out the possibilities.

Back to 8th grade English class. In the room there are probably 5 rows with six desks in each row, so 30 possible students. I do not believe we had 30, more like 24. On this particular day I was in my chair in the front row (I’ve always been one to sit in front in a class) with my friend Bonnie behind me and a girl I didn’t really know, Alice, behind her.

We are working independently on something and Mr. Albrecht, our teacher, doesn’t care if we are talking to one another. So I’m working on my project and can hear Bonnie and Alice chatting away. The two of them, who had only recently met in that class, had taken an instant liking to one another and were becoming fast friends.

Then one of them, I think Alice, asks Bonnie her birth date. To which Bonnie replies, “August First.” Alice squeals and says, “No way. My birthday is also August First.”

By now, they have my full attention. I turn around and reply, “You’re not going to believe this, but my birthday is also August First!”

“No it’s not!” Bonnie objects, “You’re joking. You’re just saying that because Alice and I the same birthday.”

I shake my head and say, “No, it really is August First.”

The debate continues for several minutes as they simply do not believe me. Finally we all agree to bring in copies of our birth certificates to prove it.

The next class day I had mine in hand and eagerly awaited the moment when I would show them I did, in fact, share the same birthday (and in this case, year) as the other two.

Pages 8 and 9 of the Wilson Junior High annual. Birthday buddies and me have checkmarks on our ‘lovely?’ photos.

We huddle together at the end of class and each produce our documents. Bonnie and Alice shake their heads in disbelief as they examine my certificate. Yes, all three of us were born on August First of the same year. It turned out, however, that I was the oldest of the trio having arrived a mere 43 minutes after midnight to make the cut.

María Laura, María Emilia and María Eugenia Fernández Roussee (born 5 July 1960). The triplets are a well known musical group in Argentina

It was a rather amazing coincidence. In all the years since I’ve never heard of another situation like it. In my Google explorations to calculate the odds of such a thing happening, it was nearly impossible to make the search engine understand what I was asking. So, all you readers out there, what ARE the odds of three random people in a group of 24 sharing the same birthday and year?

Yes, I do personally know two sets of triplets… and for the purpose of the Birthday Paradox those don’t count.

I think it is a rather narrow probability and that maybe, our little unrelated trio, defied the odds. It truly is a paradox.

A few links:

2 thoughts on “The Birthday Paradox

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