Small blank identity tags grouped in a bowl with a few matching color pairs. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
Specify the identifier space and generator
A code drawn uniformly from d possible values behaves like a birthday bucket. If each code is generated independently with replacement, the chance that at least two of n codes collide is 1 minus the probability that every draw lands in a new bucket. The same mathematics works for a toy space of 365 codes or a classroom activity with 1,000 possible labels.
Distinguish any collision from one target hit
For d = 365 and n = 23, the any-pair collision probability is about 50.7297%. A newly drawn code matching one specified existing code instead has probability 1/365. Matching any of m distinct occupied values has probability m/365. These are different events with different denominators. Do not use an any-pair threshold as a claim about one targeted guess.
Use the square-root scale carefully
When n is small compared with d, a useful approximation is P(collision) ≈ 1−exp[−n(n−1)/(2d)]. Setting that probability near one-half gives a group size around √(2d ln 2). The Birthday Paradox Calculator uses the finite-bucket product rather than relying on this approximation and limits the bucket count to one million. It is intended for educational and small-space planning examples, not evaluation of cryptographic hash security.
A duplicate check changes the process
If your system detects a duplicate and redraws before accepting an identifier, the accepted identifiers are no longer independent draws with replacement. The collision formula then describes attempted draws, not duplicate IDs left in the database. Nonuniform generators, correlated seeds and reserved values also change the model. Record the effective space, the draw mechanism and whether rejected attempts count. The companion birthday-question guide explains the any-pair distinction without the software setting. For selecting a subset without repetition, use the Combination Calculator instead of a collision calculation.
Check the model behind your result
Use the Birthday Paradox Calculator for the supported calculation and review its stated assumptions before interpreting the output. Find related tools in the Statistics category, or browse the GreenCalc homepage for other calculation topics.
By GreenCalc Editorial Team · Updated 2026-10-26