Birthday Paradox & Cryptographic Hash Collision Probability Calculator

Calculate collision probability among k items across N buckets (e.g., 365 days) and 50% threshold √(2N·ln 2).

Probability of At Least One Collision
50 %
Unique Pairs Compared k(k−1)/2 253 pairs
Probability of Zero Collisions 50 %
50% Collision Threshold (1.177·√N) 23 items
Inputs
⚡ Scenarios:
k items
Number of people or generated IDs (k).
N buckets
Total possible buckets/days N (365 for birthdays).
Results & Graph
Probability of At Least One Collision
50 %
Unique Pairs Compared k(k−1)/2
253 pairs
Probability of Zero Collisions
50 %
50% Collision Threshold (1.177·√N)
23 items

Probability of At Least One Collision

Graph 1 • Birthday Paradox Collision • Domain Architecture & Live Telemetry Maps Sample Group Size K, Total Buckets N through Birthday Paradox Collision Formula.
Probability of At Least One Co: 50 % Birthday Paradox Collision • Invariants & Metrics Probability of At Least One 50 % Unique Pairs Compared k(k−1) 253 pairs Probability of Zero Collisio 50 % 50% Collision Threshold (1.1 23 items
Graph 2 • Probability of At Least One Collision Across Sample Group Size K Sweep (40% → 160%) Evaluates P(collision) ≈ 1 − exp( −k(k − 1) / (2N) ); k_50% ≈ 1.1774 × √N across 7 operating points (current input highlighted).
0 % 46.13 % 92.26 % 9.39 9.20 22.07 13.80 34.24 18.40 50 23 64.50 27.60 74.31 32.20 83.87 36.80
Graph 3 • Joint Sensitivity: Sample Group Size K × Total Buckets N Compares Probability of At Least One Collision when Total Buckets N is shifted +25% (solid) vs. −25% (dashed).
0 % 43.76 % 87.51 % --- Total Buckets N @ 75% 23.14 -30% 34.08 -15% 42.58 +0% 50.97 +15% 61.48 +30%
Summary: Among 23 items (253 unique pairs) across 365 buckets, collision probability is 50% (50% collision threshold occurs at k = 23).
Summary: Among 23 items (253 unique pairs) across 365 buckets, collision probability is 50% (50% collision threshold occurs at k = 23).
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Formula & Mathematical Methodology: Birthday Paradox Collision Formula

P(collision) ≈ 1 − exp( −kk − 12N )
k50% ≈ 1.1774 × √N
P(collision) ≈ 1 − exp( −k(k − 1) / (2N) ); k_50% ≈ 1.1774 × √N
  • sampleGroupSizeK: Sample Group Size K (k items)
  • totalBucketsN: Total Buckets N (N buckets)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Birthday Paradox Collision

Calculate collision probability among k items across N buckets (e.g., 365 days) and 50% threshold √(2N·ln 2).

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: P(collision) ≈ 1 − exp( −k(k − 1) / (2N) ); k_50% ≈ 1.1774 × √N.
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Birthday Paradox Collision via P(collision) ≈ 1 − exp( −k(k − 1) / (2N) ); k_50% ≈ 1.1774 × √N.

Frequently Asked Questions

How does the Birthday Paradox & Cryptographic Hash Collision Probability Calculator compute the solution?

It evaluates P(collision) ≈ 1 − exp( −k(k − 1) / (2N) ); k_50% ≈ 1.1774 × √N directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Game Theory, Ratings & Stochastic Processes

Birthday Paradox & Cryptographic Hash Collision Probability Calculator: inputs, method, and how to read the result

Calculate collision probability among k items across N buckets (e.g., 365 days) and 50% threshold √(2N·ln 2).

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Sample Group Size K: Number of people or generated IDs (k).
  • Total Buckets N: Total possible buckets/days N (365 for birthdays).

What the calculation includes

With the default values shown, Probability of At Least One Collision is 50 %. The calculation follows Birthday Paradox Collision Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Sample Group Size K from 23 k items to 25.3 k items changes Probability of At Least One Collision from 50 % to 56.041 %. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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