Probability Calculator

MathLast updated: August 17, 2026

Classical (simple) probability is the ratio of favorable outcomes to the total number of outcomes: the chance of rolling a 6 on a die is 1/6, and the chance of drawing an ace from a deck is 4/52. This calculator reports that ratio as both a fraction and a percentage, and also gives the complementary probability (the event not happening).

Binomial mode covers the case where the same experiment is repeated n times independently: exactly 3 heads in 10 coin flips, or at least one 6 in 4 rolls of a die. The calculator uses the binomial distribution to find the probability of exactly k, at most k, and at least k successes, and plots the distribution.

How many outcomes count as favorable. Rolling a 6 on a die → 1.
How many outcomes are possible in total. A die → 6, a deck of cards → 52.

Formula

Simple probability: P(A) = Favorable outcomes / Total outcomes
Complement: P(A') = 1 − P(A)
Binomial: P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ
At least k successes: P(X ≥ k) = 1 − P(X ≤ k − 1)

Rather than adding terms one by one, it is easier to reach an 'at least one' probability through the complement: P(at least 1) = 1 − P(none). Mixing the two approaches is the most common mistake.

How to Calculate

  1. Choose the calculation type: simple probability for a single event, binomial for repeated trials.
  2. In simple mode, enter the number of favorable and total outcomes (for a die, 1 and 6).
  3. In binomial mode, enter the number of trials (n), the number of successes (k), and the percentage chance of success on a single trial.
  4. Read the probability as a percentage, a fraction, and a decimal; in binomial mode, also study the distribution table and chart.

Worked Examples

Rolling a 6 on a die

One of the 6 possible outcomes of a die roll is favorable: the probability is 1/6, that is 16.67%. The complementary probability of not rolling a 6 is 83.33%.

Probability (percentage): 16.67% · Probability (fraction): 1/6 · Probability (decimal): 0.1667

Rolling an even number

A die has three even numbers: 3 of the 6 outcomes are favorable. The fraction 3/6 reduces to 1/2; the probability is 50%, or 0.5000 as a decimal.

Probability (percentage): 50% · Probability (fraction): 1/2 · Probability (decimal): 0.5000

Exactly 2 heads in 5 coin flips

The probability of exactly 2 heads in 5 flips is C(5, 2) × 0.5⁵ = 31.25%. At most 2 heads has probability 50% and at least 2 heads 81.25%; the expected number of heads is 5 × 0.5 = 2.5.

Exactly 2 successes — P(X = 2): 31.25% · At most 2 successes — P(X ≤ 2): 50% · At least 2 successes — P(X ≥ 2): 81.25%

Frequently Asked Questions

How is probability calculated?
In the classical definition, probability is the number of favorable outcomes divided by the total number of equally likely outcomes. The result is always between 0 and 1, where 0 means impossible and 1 means certain. Multiply by 100 to express it as a percentage.
When should binomial probability be used?
Use it when the same experiment is repeated a fixed number of times (n), the trials are independent, and the probability of success (p) stays the same on every trial. Coin flips, dice rolls, and the number of defective items in quality control are typical examples.
How do you find the probability of 'at least one'?
The most practical route is the complement: P(at least 1) = 1 − P(none). For example, the probability of seeing at least one 6 in four rolls of a die is 1 − (5/6)⁴ ≈ 51.8%. Adding P(1) + P(2) + ... term by term gives the same answer but takes far longer.
What is complementary probability?
It is the probability that an event does not occur, found as 1 − P(A). An event and its complement always add up to 1 (that is, to 100%). This calculator reports the complementary probability alongside every result.
Can a probability be greater than 100%?
No. By definition a probability lies between 0 and 1 (0% and 100%). Because the number of favorable outcomes cannot exceed the total number of outcomes, the ratio can never pass 1; the calculator warns you if you enter such values.
What does expected value mean?
In the binomial distribution the expected number of successes is n × p: if the experiment is repeated many times, this is the average number of successes you will see. In 10 coin flips, for example, the expected number of heads is 10 × 0.5 = 5; deviations from 5 in a single run are completely normal.