Calculators
Expected Distinct Outcomes Calculator
Calculate expected unique outcomes, unseen outcomes, repeat draws, and coverage after independent uniform draws with replacement.
Estimate unique outcomes after a fixed number of independent draws with replacement from equally likely outcomes. Enter 0–1000000000000 draws and 1–1000000000000 outcomes. Six rolls of a fair die give about 3.99061 distinct faces on average. All processing stays in your browser.
How expected distinct outcomes are calculated
For n independent draws with replacement from d equally likely outcomes, E[distinct] = d × [1 − (1 − 1/d)^n]. Each outcome has probability (1 − 1/d)^n of never appearing; summing its complementary probability over d outcomes gives the expectation. Expected unseen outcomes = d − E[distinct], expected repeat draws = n − E[distinct], and expected coverage = E[distinct]/d. Six rolls of a fair six-sided die give 3.99061213992 distinct faces on average and 2.00938786008 repeat rolls. A repeat is a draw whose outcome appeared previously, not a count of matching pairs. Zero draws gives zero distinct outcomes; one possible outcome is seen after the first draw. These are averages over repeated experiments, not guaranteed counts or the probability of complete coverage. Outcomes must be uniform and draws independent with replacement; sampling without replacement needs a different model. Enter whole numbers without separators or scientific notation: 0 to 1000000000000 draws and 1 to 1000000000000 possible outcomes. Log1p and expm1 preserve small occupancy probabilities. Browser floating-point results display up to 12 significant digits; very small repeat or unseen estimates may round to zero. All inputs stay in your browser.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
How many distinct die faces appear in six rolls on average?
For a fair six-sided die, 6 × [1 − (5/6)^6] is about 3.99061213992 distinct faces.
Does expected coverage give the probability of collecting everything?
No. Coverage is the expected fraction of outcomes seen. It is not the probability that all outcomes have appeared.
How does this differ from the coupon collector calculator?
This calculator fixes the draw count and estimates distinct outcomes. The coupon collector calculator estimates draws needed to collect every outcome.
Can outcomes have unequal probabilities?
This calculator assumes equally likely outcomes and independent draws with replacement. Unequal probabilities or sampling without replacement require another model.