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Hash Collision Probability (Birthday Paradox Approximation)

Estimates the probability that at least two of n randomly hashed items collide in a hash table with m possible slots.

Written by the Vorzep Team

Computer ScienceData StructuresHashing

Birthday Paradox CalculatorP ≈ 1 − e−n²/(2m)

P = 1 − e−n²/(2·m)
Select what to solve for — enter the other two values, then click Check
Solve for:
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Collision Probability—
Low (<10%) Moderate (10–50%) High (50–80%) Very High (>80%)
P = 1 − e−n²/(2m) · n = items · m = possible values · e.g., n=23, m=365 → P≈0.507

Variables

SymbolQuantityUnit
PProbability of at least one collision
nNumber of items hashed
mNumber of possible hash values

What it means

The birthday paradox is a counter‑intuitive probability result that the chance of a collision (two items hashing to the same bucket) grows quickly with the number of items. For a hash table with m buckets, and n inserted keys, the approximate probability of at least one collision is P ≈ 1 − exp(−n²/(2m)), assuming a uniform hash function. This approximation is derived from the Poisson distribution and is accurate for n much smaller than m. It shows that even with a seemingly large number of buckets, collisions become likely as n approaches √m. For example, with m = 10⁶, a collision is already probable for n ≈ 1200. This analysis is crucial for designing hash tables: to keep collision probability low, either increase the number of buckets (lower load factor) or use a better hash function. The birthday problem also appears in cryptography (e.g., birthday attacks on hash functions) and in data deduplication. Understanding this approximation helps in tuning hash table parameters and in estimating the performance of hash‑based algorithms.

Worked example

Hash Collision Probability – Two Examples

Real‑World
Scenario 1 – 32‑bit Hash: A 32‑bit hash space has m = 2³² ≈ 4.29×10⁹ values. You store 100,000 items. What is the collision probability?
ParameterValue
n (items)100,000
m (hash space)4.29×10⁹
1P ≈ 1 − e^(-n²/(2m))
2n² = 1×10¹⁰; 2m = 8.59×10⁹; ratio ≈ 1.164
3P ≈ 1 − e^(−1.164) ≈ 1 − 0.312 = 0.688 (68.8%)
Result ≈ 69% collision probability ⚠️ Significant
Scenario 2 – 64‑bit Hash: Using a 64‑bit hash (m = 1.84×10¹⁹) with the same 100,000 items. What is the collision probability?
ParameterValue
n (items)100,000
m (hash space)1.84×10¹⁹
1n² = 1×10¹⁰; 2m = 3.69×10¹⁹; ratio = 2.71×10⁻¹⁰
2P ≈ 1 − e^(−2.71×10⁻¹⁰) ≈ 2.71×10⁻¹⁰ (0.000000027%)
Result ≈ 0.000000027% ✓ Virtually no collisions
Key insight: The birthday paradox shows collisions are more likely than expected – use at least 64‑bit hashes for non‑cryptographic applications.

Common mistakes

  • Approximation: The formula is an approximation for large m and n; for small values, the exact probability may differ.
  • Units: n is the number of keys, m is the number of buckets – both integers.
  • Assumption: Assumes a uniform hash function; if the hash is poor, collisions are more likely.
  • Complement probability: The formula gives the probability of at least one collision; the probability of no collision is e^(−n²/(2m)).
  • n²/(2m): If this product is large, the probability approaches 1; for small values, it’s close to n²/(2m).

Applications

The Birthday paradox approximation for hash collisions, P ≈ 1 − e^(−n²/(2m)), estimates the probability of at least one collision when storing n keys in m buckets. This probability grows quickly with n, making it a critical consideration in hash table design, password security, and cryptography. Engineers use this formula to choose an appropriate table size and load factor to keep collision probability acceptably low. It also appears in the analysis of distributed hash tables, birthday attacks on cryptographic hashes, and in data deduplication. Understanding this formula helps in designing robust hash‑based systems, in assessing security risks, and in balancing performance with memory usage.

  • Hash table sizing and performance tuning
  • Cryptography – analysis of collision attacks
  • Distributed hash tables (DHTs)
  • Data deduplication and similarity detection
  • Randomised algorithms and probabilistic analysis