Birthday Paradox Calculator
Calculate the probability of at least two people sharing a birthday in a group
Birthday Paradox Calculator
Enter the size of the group (1-400 people)
Choose whether to include leap years in calculations
Birthday Paradox Results
Formula: P(at least one match) = 1 - P(no matches)
P(no matches): (365/365) × (364/365) × ... × ((365-n+1)/365)
Days in year: 365 days
Probability Analysis
Famous Examples
The Classic Example
23 people: 50.73% probability
30 people: 70.63% probability
50 people: 97.04% probability
70 people: 99.92% probability
Real-World Applications
• Cryptography (birthday attack)
• Hash collision detection
• Quality control sampling
• Network protocol analysis
Quick Reference
Key Insights
Only 23 people needed for 50% probability
70 people gives 99.9% probability
Probability grows exponentially
Based on comparing all possible pairs
Understanding the Birthday Paradox
What is the Birthday Paradox?
The birthday paradox is a famous probability problem that demonstrates how our intuition about probability can be misleading. It asks: what's the probability that at least two people in a group share the same birthday?
Why is it Surprising?
- •Only 23 people needed for 50% probability
- •Much fewer than the 183 people you might expect
- •Growth is exponential, not linear
- •Number of pairs grows as n(n-1)/2
The Mathematics
P(no match) = (365/365) × (364/365) × ... × ((365-n+1)/365)
P(match) = 1 - P(no match)
Why It Works
- 1st person: Any birthday (365/365 = 100%)
- 2nd person: Must avoid 1st (364/365 ≈ 99.7%)
- 3rd person: Must avoid first 2 (363/365 ≈ 99.5%)
- And so on... Probabilities multiply
Key insight: We're not looking for a specific match, but any match among all possible pairs!