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Birthday Paradox Calculator

Math Calculators

Shared-birthday probability and people needed

Quick Answer

Birthday Paradox Calculator: Shared-birthday probability and people needed. Birthday Paradox Calculator uses standard math formulas to compute accurate results instantly.

people
P(at least two share a birthday) with 23 people
50.73%
P(no match) = 49.27%

Steps

  1. 1P(no shared birthday among n people) = 365/365 × 364/365 × … × (365−n+1)/365
  2. 2n = 23 → P(no match) = 49.2703%
  3. 3P(at least one match) = 1 − P(no match)
PeopleSceneProbability
10Small group11.69%
23Standard class50.73%
30Large class70.63%
50Large group97.04%
60Wedding hall99.41%
  • Assumes 365 uniformly distributed birthdays (leap day ignored); the counter-intuitive "23 people → 50.7%" is why this is called a paradox.
  • Real birthday distribution is not uniform (seasonal peaks), which makes the true collision probability slightly HIGHER than the theoretical value.
  • Same math underpins hash collision analysis (birthday attack) in cryptography.

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How to Use

  1. Enter values or expressions: type directly into the Birthday Paradox Calculator — keyboard friendly.
  2. Instant results: everything computes live as you type, no button needed.
  3. Follow the steps: the result panel shows full working and formula substitution.
  4. Copy or recompute: one-click copy; change any input to recalculate instantly.

Key Points

  • Birthday Paradox Calculator is a free online math tool — open and use, no sign-up.
  • All calculations run locally in your browser; your data never leaves the device.
  • Key capability: Shared-birthday probability and people needed.
  • Results come with formulas and reference tables so you can verify every number.

Formula & How It Works

Birthday Paradox Calculator formula

Birthday Paradox Calculator uses standard math formulas to compute accurate results instantly.

FAQ

How do I use the Birthday Paradox Calculator?
Open the page and fill in the input fields — results update automatically. Shared-birthday probability and people needed.
Are the results accurate?
Yes. All calculations use standard mathematical formulas and are verified for accuracy.
Is it free? Do I need to sign up?
Completely free. No registration, login or download required — just open and use.
Does it work on mobile?
Yes. The responsive design works on phones, tablets and desktops.
Is my data saved?
No. All calculations run locally in your browser. Nothing is uploaded or stored.

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Source & Updates

Standard mathematical formulas. All logic verified. Last updated: 2025-07

Last updated:

Formulas and content on this page are reviewed by the YiJianSuan team. For reference only — official sources prevail.

Maintained and periodically reviewed by the YiJianSuan editorial team — review process in ourEditorial Policy.

Cite this page: YiJianSuan. Birthday Paradox Calculator[EB/OL]. https://www.yijiansuan.com/en/c/math/birthday-paradox

Results provided by this tool are for reference only and do not constitute professional advice.