# A birthday problem

In our family, February and September are known as “birthday months”. The two months are packed with birthdays—which means juggling the budget to buy presents and deferring some other expenses.

There’s September coming up soon, I was talking to my son about “birthday expenses” and he said, “If any of your friends has invited 25 guests to the party, there’s a good chance that you will come across two people sharing a birthday.” In math, it’s called the “Birthday Paradox” or “Birthday Problem”.

Let me explain: You go to a party and find that there are some 25-odd people in the room. What are the chances that two of the guests have the same birthday (by birthday, I mean day and month, not year)? I would say the chances are pretty slim—considering that there are 365 days/birthdays in a year (ignore the leap year) and there are only 25 people in the room. But, according to the Birthday Problem, as defined on Wikipedia, “in a random group of 23 people, there is about a 50% chance that two people have the same birthday”.

I tried it out in a class I teach. There were 32 students and I started with, “All those born in January please raise your hands.” And yes, two students shared a birthday in February. I didn’t go to the next month.

Imagine that a guest arrives early at a party and there’s just one more person in the room. What is the probability that the two have the same birthday? “Just 0.27%,” according to the Birthday Problem. Increase the number of people in the room to 23 and there’s a 50:50 likelihood that two will have the same birthday (I use the words odds, probability and chance in a very broad sense, in a layperson’s language to mean likelihood).

Now this is where it becomes even more interesting: “The probability goes up to 99.9% if there are 70 people in the room.” Think again: 70 people in the room and a near-100% chance of finding two people with the same birthday. That it climbs to 100% with 367 people is easy to understand since there are only 366 possible birthdays, including 29 February.

I wouldn’t be surprised if I came across two people sharing a birthday in a large party of, say, 100 guests. I would call it coincidence; these things happen. But two people sharing a birthday in a gathering of just 23? Coincidence?

But it’s not; it’s pure math, a complicated equation. There’s a very good YouTube video, “It’s Okay To Be Smart”. The Birthday Problem says don’t look at the number in the room but at the possibilities of people sharing their birthday. It says there is a difference between “you sharing your birthday with someone” and “two people in the room sharing a birthday”. They are different sets of equations. The Birthday Problem is about the latter.

Explains *Scientific American* in an article: “If a person compares his or her birthday with the birthdays of the other people it would make for only 22 comparisons—only 22 chances for people to share the same birthday. But when all 23 birthdays are compared against each other, it makes for much more than 22 comparisons.”

During the 2014 World Cup in Brazil, there were 32 squads, each of 23 players. What are the odds that two players from one team shared a birthday? *BBC Magazine* tested the Birthday Problem: “It turns out there are indeed 16 teams with at least one shared birthday—50% of the total. Five of those teams, in fact, have two pairs of birthdays.”

My son told me an interesting story. When you buy liquor from a grocery store in the US, you have to show an ID to prove that you are over a certain age. He said that recently, when he showed his ID while paying the bill, the person at the cash counter looked at it and said, “Oh, that’s my birthday, too.” And this happened not once but twice. The answer lies in the Birthday Problem.

*Shekhar Bhatia is a science buff and a geek at heart.*