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STATISTICS CALCULATOR

Probability Calculator

Calculate a chance from equally likely outcomes or model exact, cumulative, and expected results across independent binomial trials.

Calculator

Calculation mode
%
YOUR RESULTS
Probability33.333333% %
Complement
66.666667% %
Odds / exact probability
1:2
Expected successes
4
At most selected successes
33.333333% %
At least selected successes
33.333333% %

A single trial has 2 favorable outcomes among 6 equally likely outcomes. At-most and at-least equal the single-trial probability in this mode.

Understanding your result

Single-event mode divides favorable outcomes by all equally likely outcomes. Binomial mode calculates exactly k successes and cumulative probabilities at most and at least k.

If you have a list of observed outcomes to summarize, the Mean, Median & Mode Calculator describes their center and range rather than their theoretical likelihood.

The formula

P(A) = favorable ÷ total; P(X = k) = C(n,k)p^k(1−p)^(n−k)

The binomial formula counts the arrangements of k successes, then multiplies by the probability of each arrangement. Cumulative results add exact probabilities across the relevant range.

Two heads in four fair coin tosses

With four independent trials and a 50% success chance, exactly two successes has probability 6 × 0.5⁴ = 37.5%. At most two and at least two both equal 68.75%.

How to use this calculator

  1. Choose equally likely outcomes or binomial trials.
  2. Enter the outcome counts, or the success rate, trials, and selected successes.
  3. Calculate exact probability, complement, expectation, and cumulative results.

Binomial assumptions matter

A binomial model uses a fixed number of independent trials with only success or failure on each trial.

The success probability must stay constant. Sampling without replacement from a small group can violate independence and require a different model.

Expected value is a long-run average

Multiplying trials by the success probability gives the expected number of successes across repeated experiments.

An expected value need not be a possible outcome in one experiment; three trials at 50% have an expectation of 1.5 successes.

Assumptions & limitations

What this calculation assumes

  • Single outcomes are equally likely.
  • Binomial trials are independent with a constant success probability.
  • Cumulative probabilities include the selected success count.

What to keep in mind

  • The binomial model does not cover changing probabilities or dependent trials.
  • Results use floating-point approximations.
  • Probability estimates do not guarantee an outcome.

Common questions

What is the complement of an event?

It is the probability that the event does not occur, calculated as one minus the event probability.

What does at least k include?

It includes exactly k successes and every larger possible success count.

Can probability exceed 100%?

No. Valid probability lies from 0% to 100%, inclusive.

Sources & further reading