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Category Statistics & Probability

Group Size (n)

Range: 2 – 10,000 people

Days in Year (d) Default 365. Use 366 for leap year.

Presets:

Adjust the group size and press Calculate

Probability of Shared Birthday —

Probability of NO Shared Birthday —

Number of Pairs —

Approx. Odds —

Key Thresholds for d = 365

50%—people

75%—people

90%—people

99%—people

99.9%—people

100%—people

Step-by-Step Probability Table

Person #	P(no match so far)	P(≥1 match)

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About

The Birthday Paradox states that in a group of just 23 people, the probability of at least two sharing a birthday exceeds 50%. This result is counterintuitive because humans compare themselves against the group (1⁄365) rather than counting all n(n − 1)2 pairwise comparisons. The paradox is not a true paradox but a failure of probabilistic intuition. Misunderstanding this principle leads to flawed collision estimates in hash functions, flawed cryptographic nonce generation, and underestimation of duplicate risk in datasets. This calculator computes exact probabilities using the complement method with logarithmic accumulation to maintain numerical precision for groups up to 10,000. It assumes uniformly distributed birthdays across d days and ignores leap years by default, though a custom day count is supported.

Formulas

The probability that all n people in a group have distinct birthdays across d possible days is computed as:

P(no match) = n−1∏k=0 d − kd = d!dⁿ ⋅ (d − n)!

The probability of at least one shared birthday is the complement:

P(match) = 1 − P(no match)

For numerical stability with large n, the logarithmic form is used:

ln P(no match) = n−1∑k=0 ln(1 − kd)

The well-known approximation using the Taylor expansion ln(1 − x) ≈ −x yields:

P(match) ≈ 1 − e^{−n(n−1)2d}

Where: n = number of people in the group, d = number of possible equally-likely birthdays (default 365), k = iteration index, P(match) = probability that at least two people share a birthday.

Reference Data

Group Size (n)	Probability of Shared Birthday	Odds (approx.)	Notable Context
2	0.274%	1 in 365	Single pair comparison
5	2.71%	1 in 37	Small team
10	11.69%	1 in 9	Dinner party
15	25.29%	1 in 4	Small classroom
20	41.14%	2 in 5	Tutorial group
23	50.73%	Better than even	The classic threshold
30	70.63%	7 in 10	School class
40	89.12%	9 in 10	Large lecture section
50	97.04%	33 in 34	Office floor
57	99.01%	100 in 101	99% threshold
70	99.92%	1,249 in 1,250	Near certainty
100	99.99997%	3.3M in 3.3M+1	Lecture hall
183	> 99.999...%	Pigeonhole imminent	Half of 365
366	100%	Guaranteed	Pigeonhole Principle
All values assume d = 365 equally likely days (no leap year).

Frequently Asked Questions

The common mistake is framing the question as "what is the chance someone shares MY birthday" (which is roughly n/365). The actual question counts ALL pairwise comparisons. With 23 people there are 23 × 22 / 2 = 253 distinct pairs. Each pair has a 1/365 chance of matching, and these overlapping events compound rapidly. The quadratic growth of pairs versus the linear growth of people is the core of the paradox.

Real-world birthdays are not uniformly distributed. Studies show clustering in September (in the Northern Hemisphere) and lower birth rates on holidays. Non-uniform distributions actually increase collision probability compared to the uniform model. Therefore, the calculator provides a conservative lower bound. For 23 people, the true probability with real-world data is slightly above 50.73%.

When n > d, the Pigeonhole Principle guarantees at least one collision. With d = 365, any group of 366 or more people must contain at least one shared birthday with probability exactly 1.0 (100%). The calculator correctly returns this for any n ≥ d + 1.

The Birthday Attack exploits this principle to find hash collisions. A hash function with an output of b bits has 2^b possible values. By the birthday bound, approximately 2^(b/2) random inputs are needed to find a collision with ~50% probability. This is why SHA-256 (256-bit output) provides 128-bit collision resistance rather than 256-bit. The calculator generalizes this by allowing custom values for d (possible outcomes).

Direct multiplication of terms like (364/365) × (363/365) × ... produces values extremely close to zero for large n. Floating-point arithmetic (IEEE 754 double precision) has about 15-17 significant digits. Multiplying hundreds of terms near 1.0 accumulates rounding error and can underflow to exactly 0. Summing logarithms converts multiplication to addition, preserving precision. The final result is recovered via exponentiation: P(no match) = exp(Σ ln(1 − k/d)).

Yes. The "Days in Year" parameter accepts any integer from 2 to 100,000. This generalizes the birthday problem to any collision scenario: duplicate lottery numbers, hash collisions, or matching identifiers in a database. For example, setting d = 52 models the chance of two people in a group sharing the same birth week.