Birthday Paradox Calculator
Discover the surprisingly high probability of shared birthdays in a group
The Birthday Paradox (also known as the Birthday Problem) refers to the counterintuitive fact that in a group of just 23 people, there is about a 50% chance that at least two people share the same birthday.
This is considered a paradox not because it involves a logical contradiction, but because the result is surprisingly high for most people's intuition. The actual probability reaches 99.9% with just 70 people.
Key assumptions in the classic version of the problem include:
- Birthdays are distributed uniformly throughout the year
- Each year has 365 days (leap years are ignored)
- Births are independent events
Mathematical Foundation
The probability is calculated using the following steps:
- First, calculate the probability that all birthdays are different
- For n people: P(no match) = (365/365) × (364/365) × ... × (365-n+1)/365
- Then, the probability of at least one match is: P(match) = 1 - P(no match)
Notable Probabilities
- 23 people: ~50.7% chance
- 30 people: ~70.6% chance
- 50 people: ~97.0% chance
- 60 people: ~99.4% chance
- 70 people: ~99.9% chance
Our calculator provides two different approaches:
- Theoretical Probability: Uses the exact mathematical formula to calculate the true probability
- Simulated Probability: Runs multiple trials to simulate real-world scenarios and demonstrate the theoretical probability in practice
The simulation helps visualize how the theoretical probability manifests in practice and can help build intuition about why the probability is higher than most people expect.
The Birthday Paradox has practical applications in various fields:
- Cryptography: Used in analyzing hash function collision probabilities
- Computer Science: Helps in understanding collision detection in hash tables
- Digital Security: Important in analyzing digital signature schemes
- Quality Control: Used in testing randomness of number generators
Why is the probability so much higher than expected?
The high probability comes from the number of possible pairs of people in the group, which grows much faster than the number of people. With n people, there are n(n-1)/2 possible pairs, each with a chance of sharing a birthday.
Does the specific day of the year matter?
In the classic version of the problem, all days are assumed equally likely. In reality, birth rates vary throughout the year, which would actually increase the probability of shared birthdays.
Why do you need so many simulation trials?
More trials provide a more accurate approximation of the true probability. With fewer trials, the simulated result might deviate significantly from the theoretical probability due to random chance.