Birthday Paradox Simulator
Explore how quickly shared birthdays become likely
Educational Tool Only: This simulator uses purely random numbers to demonstrate the birthday paradox. No real birthday data is used, collected, or stored. All birthdays shown are simulated for educational purposes only.
Explore the Birthday Paradox
Use this simulator to discover how likely it is that at least two people in a room share a birthday. The result is surprisingly counterintuitive!
Adjust group size
See how probability changes from small to large groups
Run simulations
Compare theory with Monte Carlo experiments
This tool uses purely simulated random numbers for educational purposes. No real birthday data is used or collected.
Understanding the Birthday Paradox
What is the Birthday Paradox?
The birthday paradox (or birthday problem) asks: How many people need to be in a room before there's a 50% chance that at least two share a birthday? The answer - just 23 - is much lower than most people expect. It's called a "paradox" because it defies our intuition about probability.
The Mathematics
The key formula calculates the probability that NO two people share a birthday: P(no match) = (365/365) × (364/365) × (363/365) × ... For n people, we multiply n terms. Then P(at least one match) = 1 - P(no match). This formula shows why matches become likely so quickly.
Why It Grows So Fast
The secret is in the number of pairs. With n people, there are n×(n-1)/2 unique pairs. At 23 people, that's 253 pairs to check! Our brains naturally think "will someone match MY birthday?" but the question is really "will ANY pair match?" - a much more likely event.
Simplifying Assumptions
This model assumes: (1) all 365 days are equally likely, (2) no leap years, (3) each person's birthday is independent. Real birthdays aren't perfectly uniform - some months are more common. But even with these simplifications, the paradox powerfully illustrates how our intuition fails with combinatorics.
Key Milestones
23
people for 50% probability
57
people for 99% probability
70
people for 99.9% probability
These numbers assume a 365-day year with uniform distribution. Interestingly, at 366+ people (more than the number of days), the probability becomes 100% by the pigeonhole principle - there simply aren't enough unique days for everyone!
Frequently Asked Questions
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Disclaimer: This simulator is for educational and entertainment purposes only. It uses random number generation to demonstrate probability concepts. No real personal birthday data is involved. For questions about probability and statistics, consult educational resources or qualified instructors.