Birthday Paradox

By Math Is Fun, July 2, 2023

Son’s birthday was approaching, and as we sat together, I decided to share with him a mind-boggling concept known as the “Birthday Paradox.” I knew it would be a fun and intriguing way to introduce him to the world of probability.

“Son,” I began, “did you know that in a room with just 23 people, there’s a high chance that two of them share the same birthday?”

Son’s eyes widened with curiosity. “Really? But there are 365 days in a year. How is that possible?”

I chuckled, excited to delve into the intricacies of this fascinating phenomenon. “Well, take a guess. What do you think the chances are that two of them share the same birthday?”

Son thought for a moment before tentatively guessing, “Maybe around 10 percent? 23 divided by 365 is only 6%”

“In a room with just 23 people, the chances of two people having the same birthday is about 50.7 percent!”

Son’s eyes widened in amazement. “That’s more than half! So it’s more likely than not that we have two people with the same birthday! But how does that work?”

I leaned closer, eager to unravel the mystery. “It’s called the Birthday Paradox. It may seem counterintuitive, but the key lies in the number of possible pairs we can make among the people in the room. You see, each person’s birthday can be paired with everyone else’s. So, for 23 people, we have 23 * 22 / 2 pairs, which is 253 pairs.”

Son’s face lit up with understanding. “Oh, I get it! So, there are many pairs, and each pair has a chance of having the same birthday.”

“Exactly!” I exclaimed, proud of his grasp of the concept. “And when you consider that each pair has a 1/365 chance of matching birthdays, the probability of at least one pair having the same birthday increases as the number of pairs grows.”

Son’s excitement grew as he began to grasp the implications. “So, if we had even more people, the chances would be even higher, right?”

I nodded, eager to explore further. “Absolutely! In fact, with just 60 people, the probability jumps to 99.9 percent! It’s a captivating illustration of how probabilities can defy our initial intuitions.”

Son’s mind buzzed with the implications. “That’s incredible! It’s like a hidden pattern in numbers, waiting to be discovered.”

I smiled, thrilled by his enthusiasm for the beauty of mathematics. “Indeed, son. The Birthday Paradox is a remarkable example of how mathematics can surprise and amaze us. It reminds us that even in seemingly random situations, patterns and probabilities emerge.”

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