Imagine walking into a room with 22 strangers. What are the odds that two of you share the same birthday? Most people would guess low, after all, there are 365 days in a year. But surprisingly, the probability is over 50%. This counterintuitive result is known as the Birthday Paradox, and it’s a classic example of how human intuition often fails when it comes to probability.
What Is the Birthday Paradox?
The Birthday Paradox (also called the Birthday Problem) refers to the surprising fact that in a group of just 23 people, there's a greater than 50% chance that at least two people share a birthday. With 50 people, that probability jumps to 97%, and by 60, it’s nearly certain at 99.4%.
This isn’t about someone sharing your birthday; it’s about any two people in the group sharing a birthday. That distinction is key.
Why Does It Work?
The paradox arises from the way probabilities compound. In a group of 23 people, there are 253 unique pairs (calculated as ( \frac{23 \cdot 22}{2} )). Each pair has a chance of matching birthdays, and those chances add up quickly.
Here’s a simplified breakdown:
- The probability that two people don’t share a birthday is ` \frac{364}{365}`.
- For three people, the chance that none share a birthday is `\frac{364}{365} \cdot \frac{363}{365}`.
- Continue this pattern for 23 people, and the probability that no one shares a birthday drops below 50%.
🧠Why Is It Counterintuitive?
Humans tend to think linearly. We focus on our own birthday and compare it to others, so we expect low odds. But the paradox involves many-to-many comparisons, not one-to-many. That’s why the probability grows so fast.
Try It Yourself
Next time you’re in a group of 23 or more, ask everyone their birthday. You might be surprised how often a match pops up!
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