Free · Probability and statistical models

Birthday Pair Collision Approximation

Estimate a shared birthday using an approximation that treats all person pairs independently.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

Two people create one pair. Under 365 equally likely dates their collision chance is 1/365.

p ≈ 1 − (1 − 1/D)^(n(n−1)/2)

Worked example

Enter these known values and leave the other values blank.

Number of people
2
Number of pairs
1
Probability of sharing a birthday
0.2739726027 %

Assumptions and limitations

  • This is the captured independent-pairs approximation, not the exact birthday-paradox probability. Pair collision events are dependent when three or more people are present.
  • Birthdays are modeled as equally likely. The 365.25 option is an approximation to leap-year averaging.
  • Use nonnegative whole person counts. For n greater than the number of possible dates the exact probability is 1, even when this approximation is below 1.

Common questions

What should I check before using this result?

This is the captured independent-pairs approximation, not the exact birthday-paradox probability. Pair collision events are dependent when three or more people are present.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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