Every time I meet a Taurus, I get so excited, like I’m meeting my missing sister or brother for the first time. Sadly, I have only been acquainted with one person who shares the same birthday as me, and even this coincidence is not a matter of luck, because we were born in the same hospital. So for the past sixteen years, I have believed that it is almost impossible to find someone sharing the same birthday with you in a random circumstance. However, an essential branch of mathematics, probability, has a different story to tell.
The Birthday Paradox
Imagine your best friend invited you, alongside thirty other people from your grade, to their birthday party. After finishing the cake and beverages and with nothing more to do, everyone starts to find who shares the same birthday. Surprisingly, two boys who have never talked to each other were born on the same date! Now, you are trying to find how many people are needed to make sure at least two of them share the same birthday.
Intuitively, the answer would be 365/2 ≈ 182. If you got this number, then you seriously underestimate the growth of probability along with growth of samples. According to Jim Frost, a mathematician in the American Society of Quality's Statistics Digest, “The number of possible pairings increases exponentially with group size. And humans are terrible when it comes to comprehending exponential growth.”
So what’s the right answer? And how does the math work?
Let’s assume there is n people in the party and we need to pair them up to compare their birthday. The first person has n-1 counterparts, the second one has n-2, because he has already been paired with the first person. So the number of different pairs would be (n-1)+(n-2)+(n-3)+(n-4)+......+1+0 . With the formula we learned in the unit of arithmetic sequence, nth sum of the pairs can be expressed as
(If you have learned combination and permutation, you must be familiar with this formula.)
The chance of two random people not being birthday twins is:
When you have two pairs, the chance that none of them share birthdays is:
So chance that one pair shares birthday is:
Pretty low and makes a lot of sense, huh? But watch how fast an exponential function can grow. Here we have the function of the possibility than a pair of birthday twins exists in n people:
The graph looks like this. Because percentage is too small to be seen, I multiply the function by 100, and the possibility can be expressed as y%.
We all know that the percentage would eventually reach 100% when n=366. The interesting thing is how rapidly this function grows. When the percentage reaches 50%, n ≈ 23, which means the possibility of two people sharing the same birthday is 50% in 23 people.
Yes, only 23 people.
So what can we learn from this function? First, never underestimate the power of exponential growth. Second, be proactive at parties because it is very likely that you would find your birthday twins!
Work Cited:
Betterexplained. (2024). Understanding the Birthday Paradox – BetterExplained. [online] Available at: https://betterexplained.com/articles/understanding-the-birthday-paradox/ [Accessed 31 Jan. 2024].
Dove, L.L. (2023). What Is the Birthday Paradox? [online] HowStuffWorks. Available at: https://science.howstuffworks.com/math-concepts/birthday-paradox.htm [Accessed 31 Jan. 2024].
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