CalcHab

Birthday Paradox Calculator & Simulator

The birthday paradox demonstrates how counter-intuitively fast the chance of matching birthdays rises. In a room of just 23 people, there is a greater than 50% chance that two share a birthday.

Combinations formula: N × (N - 1) / 2

23
Quick presets:
Theoretical Match Probability
50.73%
Критична точка переходу (>50%)
Total Pair Combinations
253

При 23 людях перевіряється 253 пар. Саме тому ймовірність збігу зростає так стрімко!

Current Random Room Sample

Випадковий екземпляр кімнати з 23 осіб

All birthdays are unique

Monte Carlo Simulation Results

Монте-Карло тестування парадоксу наживо

Probability curve by group size
N=23 → P=50.7%
50%
N = 2N = 23 (50%)N = 50 (97%)N = 80 (99.9%)

Why does our intuition fail?

People intuitively assume you need around 183 people (half of 365). The misunderstanding is thinking about matches with YOUR OWN birthday. However, the paradox considers ANY pair sharing a birthday. Among 23 people, there are 253 possible pairs to compare, making a shared birthday more likely than not (>50%)!

How to use

  1. Enter group size (number of people).
  2. Run calculation or monte-carlo trial.
  3. View exact probability that at least two people share a birthday.
  4. Vary participant count to observe the steep exponential curve.

Probability Examples

For 23 people, the probability of a shared birthday is approximately 50.7%.

For 10 people: 11.7%.

For 20 people: 41.1%.

For 30 people: 70.6%.

For 50 people: 97.0%.

Calculation Methodology

Complementary probability is used: P(no match) = (365/365) * (364/365) * ... * ((365 - n + 1)/365). The probability of at least one match is P(match) = 1 - P(no match).

Frequently Asked Questions

Why is it called a paradox?

Because human intuition underestimates the number of pairwise combinations, which scales as n*(n-1)/2.

How many people are needed for a 50% match chance?

Only 23 people are required under the 365-day assumption.

Does this apply to year or day of week?

It applies strictly to the calendar day and month, regardless of birth year or day of the week.

Are leap years considered?

The standard model assumes 365 equally likely calendar dates.

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