What is the birthday paradox?

It’s the surprisingly high probability that at least two people share a birthday in a group. With 23 people (365-day year), the collision probability is about 50.7%.

Does this tool only work for birthdays?

No. It also works for generic collisions: n draws from N equally likely outcomes (buckets), and hash spaces modeled as N = 2^b outcomes.

What probability is being calculated?

The probability of at least one collision (at least one matching pair). The tool also shows the complementary probability of no collisions (all outcomes unique).

What approximation does it use?

A common approximation is P(collision) ≈ 1 − exp(−n(n−1)/(2N)). It’s very accurate when n is small relative to N.

How accurate is the Monte Carlo simulation?

Simulation accuracy improves with more trials. The tool reports an estimated probability and a 95% confidence interval based on the simulated frequency.

Why can collisions be likely even when N is large?

The number of pairs grows as n(n−1)/2. Even if N is big, enough pairs makes a match likely.

Does this prove a hash function is broken?

No. The birthday bound describes generic collision probability in a finite output space. It’s a statistical phenomenon, not a vulnerability by itself.

Do real birthdays follow a uniform distribution?

Not perfectly. The classic paradox assumes uniform birthdays. Real-world seasonality can slightly change the probability; this tool is an idealized calculator.

What happens if n > N?

A collision is guaranteed by the pigeonhole principle, so P(collision)=1 and P(no collision)=0.

Does the share link preserve my inputs?

Yes. “Share Calculation Link” creates a URL that encodes your current inputs so the page can restore and re-run the same setup.

Birthday Paradox Calculator — Collision Probability (2025)

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Enter your setup

This tool computes the probability of at least one collision among n draws from N equally likely outcomes. In “birthday mode” we use N = 365 (or 366 with leap day).

Mode

Group size (n)

Birthday year length

Target probability (optional)

Plot up to n

Monte Carlo simulation

Simulation trials

Seed (optional)

Quick presets:

Results

P(collision) — at least one match

—

P(no collision) — all unique

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Approx. P(collision)

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Expected matching pairs

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Target probability estimate:

Estimated n for 50% collision probability: —

Monte Carlo simulation

Simulated P(collision): —

Quick thresholds (approx.)

Target P(collision)	Estimated n
10%	—
25%	—
50%	—
75%	—
90%	—
99%	—

Note: This calculator assumes outcomes are equally likely (uniform). Real-world birthdays are not perfectly uniform. Hash mode results describe the generic birthday bound and are not security advice.

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📌 Overview — Birthday Paradox & Collision Probability Calculator

The birthday paradox asks a simple question: “In a group of n people, what is the probability that at least two share the same birthday?” Even though there are 365 possible birthdays, the answer becomes “surprisingly large” much sooner than most people expect. This is why it’s called a paradox — not because it’s wrong, but because it breaks intuition.

This page is a dedicated birthday paradox calculator, but it also generalizes the idea to a more universal collision question: if you draw n items from N equally likely outcomes (also called buckets), what is the probability of a collision (at least one repeated outcome)? That generalization makes this tool useful for:

Teachers & students: probability demos, homework checking, intuition building
Computer science learners: understanding the birthday bound and collision scaling
Data & product teams: estimating duplicate risk for IDs, random assignments, or bucket sampling

What this tool shows:

Exact probability of at least one collision (when feasible)
Approximation formula that stays fast for huge N
A plot vs. group size so you can see how risk grows
Optional Monte Carlo simulation (great for classroom demos)

If you like probability tools with charts and simulation, you may also enjoy: Coin & Dice Probability Lab, Ball & Urn Probability Calculator, and the Hypergeometric Distribution Calculator.

🧮 Formula & Methodology — Exact Probability, Complement, and Approximation

The cleanest way to compute “at least one shared birthday” is to use the complement: compute the probability that all birthdays are different, then subtract from 1. This same logic works for N buckets and hash spaces.

1) Exact (uniform outcomes)

Let N be the number of possible outcomes (365 for birthdays), and n be the group size. The probability of no collision (all unique) is:

P(no collision) = (N/N) × ((N−1)/N) × ((N−2)/N) × ... × ((N−(n−1))/N)
               = ∏(k=0 to n−1) (N−k)/N

Then the probability of at least one collision is:

P(collision) = 1 − P(no collision)

If n > N, a collision is guaranteed (pigeonhole principle), so P(collision)=1.

2) Approximation (birthday bound)

When n is small compared to N, a standard approximation is:

P(collision) ≈ 1 − exp( − n(n−1) / (2N) )

This is extremely accurate for birthdays (N=365) up to the typical classroom ranges, and it remains useful for very large N (like hash spaces).

3) A quick “rule-of-thumb” scale

Collisions become likely when n reaches the order of √N. In hash terminology, this is why a b-bit hash (N = 2^b) has a collision “scale” around 2^(b/2).

Expected matches: A simple expectation often used for intuition is the expected number of matching pairs:

E[matching pairs] = C(n,2) / N = n(n−1) / (2N)

When this value is around 1, collisions are no longer “rare”.

✅ Examples — Shared Birthday Probability and Generic Collisions

Example A: Classic birthdays (N = 365)

n = 23 → P(collision) ≈ 50.73% (the famous threshold)
n = 30 → P(collision) ≈ 70.63%
n = 57 → P(collision) ≈ 99.01%

Example B: Buckets / IDs (N = 1000 outcomes)

Suppose you randomly assign 50 people into 1000 equally likely buckets:

N = 1000, n = 50 → collision probability ≈ 71%

Example C: Hash collisions (N = 2^64)

A common “birthday bound” landmark is the group size for ~50% collision probability:

64-bit space → n ≈ 5.06 × 10⁹ items for ~50% collision risk (approx.)

(This is an idealized model and does not imply anything is “broken” — it’s a generic statistical bound.)

🖼️ Infographic & Visual Guide — “Why √N Matters”

A good way to remember the birthday paradox is to look at how the number of pairs grows. In a group of n, there are n(n−1)/2 pairs. Each pair is an “opportunity” for a match. That’s why collision risk increases much faster than linear intuition suggests.

Key “birthday paradox” checkpoints (365-day year):

Group size n	P(at least one shared birthday)	Intuition note
10	≈ 11.69%	Already non-trivial
20	≈ 41.14%	Close to “coin flip”
23	≈ 50.73%	Classic paradox point
30	≈ 70.63%	More likely than not
50	≈ 97.04%	Almost certain

🎯 Use Cases — Where Collision Probability Shows Up

Classroom demos: show exact vs approximation vs simulation and discuss variance
Programming & CS: understand why collision risk scales as ~n² via pairs
Hash-space intuition: reason about collision likelihood in a finite output space (birthday bound)
Random IDs / buckets: estimate duplicates when assigning users into a limited set of categories
Product experiments: sanity-check whether a “random assignment” system could collide at scale

If you’re exploring more probability models: the Hypergeometric Calculator covers sampling without replacement with successes/failures, and the Coin & Dice Lab covers independent-trial events with exact formulas and simulation.

Assumptions & limits:

Uniform outcomes: each of the N outcomes is equally likely
Independence: each draw is independent and identically distributed
Simulation: uses browser pseudo-randomness; for very large N (e.g., 2^128) simulation is not meaningful

❓ FAQ — Birthday Paradox Calculator & Collision Probability

Why is it called a “paradox”?

Because intuition often expects you need “hundreds” of people before a shared birthday becomes likely. In reality, the number of pairs grows quickly, and at n=23 the probability is already about 50.7%.

What exactly counts as a “collision” here?

A collision means at least one repeated outcome (at least one matching pair). It does not require everyone to share the same birthday — just any match at all.

Is the exact result always computed?

For typical birthday ranges it is. For very large n (hundreds of thousands or more), the tool may rely on the approximation to stay fast, because exact products are O(n) and must be computed carefully for numerical stability.

How good is the approximation?

The approximation 1 − exp(−n(n−1)/(2N)) is excellent when n ≪ N. For birthdays it tracks the exact curve very closely across common classroom values.

Why does √N appear in so many explanations?

Collisions become likely when the number of pairs n(n−1)/2 is comparable to N. Solving n²/2 ≈ N gives n ≈ √(2N), so the “collision scale” grows like √N.

Do real birthdays follow the uniform assumption?

Not perfectly. Birth rates vary by season and geography, which can slightly change collision probabilities. This tool models the classic uniform-birthday paradox for clean intuition and math practice.

What if I include leap day (366 outcomes)?

Using 366 slightly reduces collision probability for the same n (because there are more possible outcomes). The tool lets you toggle this if you want the “365 vs 366” comparison.

What does “expected matching pairs” mean?

It’s the expected number of pairs that match: E = C(n,2)/N. It’s a helpful intuition metric: when E is around 1, collisions are no longer rare.

Does a high collision probability mean a hash function is broken?

No. The birthday bound is a generic property of finite output spaces. It helps you reason about how collision risk scales with the number of items hashed; it is not a verdict on a specific algorithm.

How many simulation trials should I run?

Start with 10,000–20,000 trials for a stable estimate. Increase trials if you want a tighter confidence interval, but note it increases runtime.

Can I export or share my calculation?

Yes. Use the action buttons to copy results, export TXT/CSV, print/save PDF, or create a share link that preserves your inputs.

Is this tool advice for security or cryptography?

No. It’s an educational calculator for probability intuition. If you’re making security decisions, consult current standards and qualified guidance.

Reviewed: December 2025 — formulas, thresholds, and interpretation guidance checked for clarity.

Important Disclaimer:
This calculator is provided for educational and informational purposes only. It assumes uniformly random outcomes and independence. Monte Carlo simulation uses browser randomness and is not suitable for security-critical validation or high-stakes decision-making.

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