How do I simulate a sampling error?

Open this page and use the live Birthday Paradox tool below — set your inputs and the simulation runs instantly in your browser using real numerics. No install, no account needed.

Yes. The simulation runs free in your browser. A one-time unlock or a Pro plan adds advanced parameters, saved presets, data import, and clean exports.

Can I use my own numbers?

Absolutely — every input is adjustable, and with data import you can drive a sampling error from your own measurements.

Use case · powered by Birthday Paradox

Birthday Paradox for a sampling error

Simulate a sampling error live in your browser. This runs the real Birthday Paradox solver — adjust the inputs, watch it respond instantly, and export the result. No install, no account.

For:Students Researchers Educators Engineers Hobbyists & Makers K-12 Students First Responders

Birthday ParadoxLive

smaller than you'd think

Controls

Presets

Group size

How many people until two share a birthday? Intuition says hundreds, but it takes just 23 for better-than-even odds, and 70 makes it near-certain. The trick is that the number of pairs grows quadratically — 23 people form 253 pairs, each a chance to match. It is why hash collisions and cryptographic birthday attacks happen far sooner than expected.

▶ Run in Python

Data Inspector

Group size23

P(shared birthday)50.7%

Pairs253

Governing equation

P (n) = 1 - i = 0 \prod n - 1 \frac{365 - i}{365} = 1 - \frac{365 !}{( 365 - n )! 36 5 ^{n}} = 50.7% (n = 23)

Reading this result: Past the tipping point: 23 people form 253 pairs, and a match is now more likely than not (50.7%) — far fewer people than the 365 intuition expects.

Runs locally in your browser — free forever. Scale to the cloud when reality gets heavy.

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About this simulation

The full Birthday Paradox tool models a sampling error with the same numerics engineers and scientists use — running entirely client-side. Change any parameter and the result updates in real time, so you can build intuition, check a design, or teach the concept without spreadsheets or installs.

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Frequently asked questions

How do I simulate a sampling error?: Open this page and use the live Birthday Paradox tool below — set your inputs and the simulation runs instantly in your browser using real numerics. No install, no account needed.
Is it free?: Yes. The simulation runs free in your browser. A one-time unlock or a Pro plan adds advanced parameters, saved presets, data import, and clean exports.
Can I use my own numbers?: Absolutely — every input is adjustable, and with data import you can drive a sampling error from your own measurements.