Simulate the birthday problem for groups of various sizes. Compare theoretical probability with empirical results across trials.
This tool answers the classic birthday-paradox question for any group size you choose: what is the chance that two people in the room share a birthday? It computes the exact theoretical probability from the counting formula, then runs your requested number of random trials, assigning each simulated person a birthday and checking for a match, so you can watch the empirical frequency converge toward the theoretical curve. With the default 365-day year, the theoretical probability crosses 50% at just 23 people, and the empirical result jumps around noticeably at low trial counts before settling down closer to that curve as trials increase.
A concrete input and expected output from the current implementation.
Input
Group size: 23 people, Trials: 1000, Days in year: 365
Expected output
Theoretical probability of a shared birthday: approximately 50.73%. Empirical probability across 1000 trials: will vary run to run, typically landing within a few percentage points of 50.73% (roughly 47%-54%), since the standard error at 1000 trials for an event near p=0.51 is about 1.6 percentage points.
With 23 people, 1 minus the product of (365-i)/365 for i=0 through 22 gives about 50.73%; the simulated (empirical) value is random by construction, so it scatters around that figure rather than matching it exactly.
For a group of n people and a chosen number of days d (the 'Days in year' setting, 365 by default), the tool multiplies (d-i)/d for i = 0 to n-1 to get the probability of zero shared birthdays, then subtracts that product from 1. This is the exact closed-form value, not an approximation, and it recomputes instantly whenever you change the group size or the days-in-year setting.
For each of the trials you request, the tool draws n independent random birthdays for the group, checks whether any two coincide, and tallies how many trials had a match. Dividing matches by trial count gives the empirical probability, shown alongside the theoretical value so you can compare how quickly it converges.
Each run draws a fresh set of random birthdays, so the empirical result is a new random sample every time, not a fixed number. Increasing the trial count reduces this variance and pulls the empirical value closer to the theoretical one, but it will essentially never land on it exactly.
No. It models birthdays as uniformly and independently distributed across the days in the year, which is the standard assumption behind the classic birthday-problem formula. If you need the collision rate under a real, non-uniform birth distribution, this tool will not produce that number.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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