Birthday Paradox Calculator

Find the probability that at least two people in a group share a birthday — the famous, counterintuitive birthday problem.

Inputs

How many people are in the room (1–365).

Usually 365; use 366 to include leap day.

Result

Probability at least two share a birthday
50.73%
23 people, 365-day year
  • People23
  • Probability of a shared birthday50.7297%
  • Probability all birthdays differ49.2703%
  • Odds1.03 : 1 in favor
Source: P(at least one shared) = 1 − ∏_{i=0}^{k−1} (d − i)/d, assuming uniform, independent birthdays.
Note — Assumes birthdays are uniformly distributed and independent and ignores leap years and seasonal birth patterns. Real-world clustering makes a match slightly more likely.

Step-by-step

  1. Probability all 23 birthdays differ = ∏(d−i)/d for i=0..22 = 49.2703%.
  2. Probability at least two match = 1 − 49.2703% = 50.73%.

How to use this calculator

  • Enter the number of people in the group.
  • Leave days at 365 (or set 366 to include leap day).
  • Read the probability that at least two people share a birthday.
  • Try 23 for the classic ~50% result, or 70 for near-certainty.

About this calculator

The birthday paradox is the surprising fact that in a group of just 23 people, there is about a 50% chance that two of them share a birthday — and with 70 people it rises above 99.9%. It feels wrong because we instinctively think about how many people share our specific birthday, but the question is whether any two people match, and the number of possible pairs grows quadratically with group size. The calculation works by finding the probability that everyone's birthday is different — multiplying 365/365 × 364/365 × 363/365 and so on — and subtracting from one. This calculator computes that probability for any group size, assuming birthdays are spread uniformly across the year.

How it works — the formula

P(no match) = ∏_{i=0}^{k−1} (d − i) / d P(at least one match) = 1 − P(no match)

The chance every birthday is distinct is the product of each successive person avoiding all earlier birthdays; one minus that is the chance of at least one collision.

Source: Wikipedia — Birthday problem

Worked examples

Example 1
23 people
Inputs:
people=23, days=365
Output:
50.73% chance of a shared birthday
Example 2
50 people
Inputs:
people=50, days=365
Output:
97.04%
Example 3
70 people
Inputs:
people=70, days=365
Output:
99.92%

Limitations

  • Assumes uniform, independent birthdays across the year.
  • Ignores leap days and real seasonal birth patterns.
  • Limited to a single shared-birthday question, not triples or specific dates.

Idealized model; real-world probabilities are marginally higher due to birth clustering.

Frequently asked

Because the answer defies intuition: most people guess you would need around 180 people for a 50% chance, but only 23 suffice. It is not a true logical paradox — just a counterintuitive result driven by the large number of possible pairings.

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