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Birthday Paradox Simulation

Simulation

Parameters

Simulated Probability:0.00%
Theoretical Probability:0.00%

Simulation Chart

How to Interpret the Results

  • The blue line shows the simulated probability as more trials are run
  • The red dashed line shows the theoretical probability
  • With 23 people, there's about a 50% chance of a shared birthday
  • The simulation helps visualize how quickly this probability increases with group size

Learn More

Understanding the Birthday Paradox

Overview

The Birthday Paradox demonstrates how our intuition about probability can be misleading. It shows that in a relatively small group of people, the probability of two people sharing a birthday is surprisingly high.

Mathematical Foundation

The probability is calculated using the complement rule:

P(shared birthday)=1P(no shared birthdays)P(\text{shared birthday}) = 1 - P(\text{no shared birthdays})

For n people:

P(no shared birthdays)=365!(365n)!×365nP(\text{no shared birthdays}) = \frac{365!}{(365-n)! \times 365^n}

Key probabilities:

  • 23 people: ~50% chance
  • 30 people: ~70% chance
  • 50 people: ~97% chance
  • 70 people: ~99.9% chance