Birthday Paradox Calculator

Online calculator to compute the birthday paradox probability

Birthday Paradox Calculator

The Birthday Paradox

The birthday paradox describes the surprisingly high probability that in a group of n people at least two share the same birthday.

Enter parameters
Number of people in the group (1-500)
Probability Results
Probability:
Paradox Properties

Surprising fact: With only23 people the probability is already above50%

23 people ?50% 30 people ?70% 50 people ?97%

Probability Curve

The probability increases quickly with the group size.
At23 people it already exceeds50%.

Birthday Paradox

? At least two same birthdays
? All birthdays different


What is the Birthday Paradox?

The birthday paradox is one of the best-known examples of counterintuitive probabilities:

  • Definition: Probability that at least two people in a group share the same birthday
  • Paradox: The probability is much higher than intuitively expected
  • Magic number: Already above50% at23 people
  • Applications: Cryptography, hashing, collision detection
  • Principle: Exponential growth of the number of pairs
  • Related: Coupon collector problem, birthday attack

Why is the probability so high?

The high probability is explained by the exponential growth of the number of pairs:

Number of pairs
  • 2 people:1 pair to compare
  • 3 people:3 pairs to compare
  • 23 people:253 pairs to compare
  • n people: n(n-1)/2 pairs
Probability milestones
  • 10 people: ?11.7% probability
  • 20 people: ?41.1% probability
  • 23 people: ?50.7% probability
  • 30 people: ?70.6% probability

Applications of the Birthday Paradox

The birthday paradox has important practical applications:

Cryptography & Security
  • Birthday attacks on hash functions
  • Collision detection in encryption
  • Vulnerability analysis of algorithms
  • Random number generator tests
Computer Science & Data Processing
  • Hash table design and optimization
  • Duplicate detection in large datasets
  • Load-balancing algorithms
  • Probabilistic data structures
Statistics & Research
  • Sample size determination
  • Random experiments and Monte Carlo methods
  • Probability education and teaching
  • Empirical validation of theories
Practical applications
  • Quality control in manufacturing
  • Scheduling and resource management
  • Risk assessment in insurance
  • Network traffic analysis

Formulas for the Birthday Paradox

Basic relation
\[P(\text{at least2 equal}) =1 - P(\text{all different})\]

Complementary probability

All different birthdays
\[P(\text{all different}) = \frac{365!}{365^n \cdot (365-n)!}\]

Probability that n birthdays are all different

Product formula
\[P(n) =1 - \frac{365}{365} \cdot \frac{364}{365} \cdot \frac{363}{365} \cdots \frac{365-n+1}{365}\]

Stepwise calculation for all different birthdays

Compact product form
\[P(n) =1 - \prod_{i=0}^{n-1} \frac{365-i}{365}\]

Product notation with index

Approximation (Stirling)
\[P(n) \approx1 - e^{-\frac{n(n-1)}{2 \cdot365}}\]

Approximation for large n

50% threshold
\[n_{50\%} \approx \sqrt{2 \ln(2) \cdot365} \approx22.49\]

Estimate for50% probability

Example calculations for the Birthday Paradox

Example1:3 people
n =3
Step1: All different birthdays
\[P(\text{all different}) = \frac{365}{365} \times \frac{364}{365} \times \frac{363}{365}\] \[=1 \times0.9973 \times0.9945 =0.9918\]
Step2: At least one match
\[P(\text{at least2 equal}) =1 -0.9918\] \[=0.0082 =0.82\%\]
Example2:23 people (The magic number)
n =23
Product calculation
\[P(\text{all different}) = \prod_{i=0}^{22} \frac{365-i}{365}\] \[= \frac{365 \times364 \times \cdots \times343}{365^{23}}\] \[\approx0.4927\]
Paradox result
\[P(\text{at least2 equal}) =1 -0.4927\] \[=0.5073 = \mathbf{50.73\%}\]
Over50% probability!
Example3:50 people
n =50
Use approximation
\[P \approx1 - e^{-\frac{50 \times49}{2 \times365}}\] \[=1 - e^{-\frac{2450}{730}}\] \[=1 - e^{-3.356} \approx1 -0.0347\]
Impressive result
\[P(\text{at least2 equal}) \approx0.9653\] \[= \mathbf{96.53\%}\]
Almost certainty!
Probability table
People Pairs Probability People Pairs Probability
104511.7%4078089.1%
1510525.3%50122597.0%
2325350.7%60177099.4%
3043570.6%70241599.9%

Mathematical foundations of the Birthday Paradox

The birthday paradox is a classic example of counterintuitive probabilities and vividly demonstrates how the human mind struggles to assess exponential growth effects. It is based on combinatorial principles and the computation of complementary probabilities.

Combinatorial basics

The paradox relies on the combinatorial analysis of possible birthday distributions:

  • Number of pairs: For n people there are \(\binom{n}{2} = \frac{n(n-1)}{2}\) pairs to compare
  • Quadratic growth: The number of comparisons grows quadratically with the number of people
  • Complementary event: It is easier to compute the probability that all birthdays are different
  • Product rule: Probabilities of independent events are multiplied
  • Normalization:365 possible birthdays assuming no leap years

Why is it paradoxical?

Most people drastically underestimate the probability for several reasons:

Cognitive biases

People think linearly rather than exponentially and overestimate the importance of365 possible days instead of focusing on the number of pairs.

Wrong analogy

Confusing with the question "How likely is someone to be born on MY birthday?" (much lower probability).

Underestimating combinatorics

The quadratic increase in pair comparisons is not intuitively captured. With23 people there are already253 pairwise comparisons.

Neglecting small probabilities

Each pair comparison has a small collision probability, but the accumulation over all pairs results in a high total probability.

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