A fair die
In the equally likely model, 1 favorable face out of 6 gives probability 1/6, or about 16.6667%. This ratio assumes all six faces have the same chance.
Calculations run locally in your browser. This tool does not upload the numbers you enter.
How to use the Probability Calculator
- 01
Select the model before entering values. Use counts for equally likely outcomes or decimal probabilities from 0 to 1 for the other models.
- 02
For a binomial calculation, enter total trials n, target successes k, and the constant probability p of success on one trial.
- 03
Calculate and review exact-count, inclusive-tail, or combined-event probabilities as appropriate. Copy or download the report.
Examples
Three heads in ten flips
With n = 10, k = 3, and p = 0.5, the binomial probability of exactly three successes is 0.1171875. The at-most-three probability is 0.171875.
A count ratio needs equal likelihood
Dividing favorable outcomes by total outcomes works when the individual outcomes are equally likely. It does not model a weighted spinner merely by counting its colored regions. The complement is one minus the event probability.
Combining independent events
For independent A and B, the joint probability is P(A)P(B). The inclusive union adds both individual probabilities and subtracts the overlap. Exactly one includes A without B and B without A. Independent does not mean mutually exclusive: two independent events can occur together.
The binomial model
A binomial count assumes a fixed number of independent trials, two possible outcomes on each trial, and the same success probability every time. Exactly k successes uses a binomial coefficient times p^k(1 − p)^(n − k). The two tails include k itself. Probabilities are accumulated numerically; extremely small values can underflow to zero.
What to know
- Binomial trials are capped at 1000. Counts in the equally likely model are capped at 1e12.
- Probability output is numerical, rounded for display, and assumes the selected model is appropriate.
Common questions
Should I enter 25 or 0.25 for 25%?
Enter 0.25. Probability fields accept values from 0 through 1 rather than whole percentages.
Can I use this for dependent events?
The two-event model assumes independence. If one event changes the likelihood of the other, provide a suitable conditional model elsewhere instead of using the independent-event product.
Does at least k include k?
Yes. At least means X ≥ k, while at most means X ≤ k. Both cumulative answers include the exact-k outcome.
Can n or k be zero?
Yes. The binomial model allows 0 ≤ k ≤ n ≤ 1000. With zero trials, zero successes has probability one.