The Birthday Paradox
How many people do you need in a room before there's a 50% chance that two people share the same birthday? The answer might surprise you!
The Birthday Paradox shows that with just 23 people, there's already a 50% chance of a shared birthday. With 70 people, the probability jumps to 99.9%!
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Shared Birthday
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Random Birthdays
Run simulation to see random birthdays
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Birthday Distribution
Days with shared birthdays
Days with single birthdays
The Mathematics
The Birthday Paradox is counterintuitive because we tend to think about comparing our birthday to others individually, rather than considering all possible pairs.
With n people, there are n(n-1)/2 possible pairs. The probability that no two people share a birthday is:
P(no shared birthday) = 365! / (365^n × (365-n)!)
Therefore, the probability of at least one shared birthday is:
P(shared birthday) = 1 - P(no shared birthday)
This formula shows why the probability grows so quickly with the number of people!
Fun Facts
- With 23 people: ~50% chance of shared birthday
- With 50 people: ~97% chance of shared birthday
- With 70 people: ~99.9% chance of shared birthday
- With 100 people: ~99.99997% chance of shared birthday
- This is why hash collisions in computer science are more common than expected!