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Simple Calculators

Probability Calculator

Calculate probability, combinations, permutations, and more — with step-by-step working

Probability Calculator

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Quick summary

Probability measures how likely an event is to occur on a scale from 0 (impossible) to 1 (certain). Combinations count unordered selections; permutations count ordered arrangements. The complement rule and addition rule help calculate probabilities of related events.

Reviewed by Simple Calculators TeamLast reviewed

How to use this calculator

  1. Enter the inputs requested by the calculator.
  2. Choose the units or options that match your situation and review the assumptions.
  3. Run or update the calculation to see the result.
  4. Review the result, breakdown, and formulas shown on the page.
  5. Check the inputs, units, dates, and limitations before saving or sharing the result.

Probability Formulas & Examples

Basic Probability

P(E)=Favorable OutcomesTotal OutcomesP(E) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}
P(E)
Probability of event E (between 0 and 1)

Worked example

Rolling a 4 on a fair die

Given
favorable = 1
total = 6
Calculation
P(4)=1/6P(4) = 1/6
Result
≈ 16.67%

Combinations C(n, r)

C(n,r)=(nr)=n!r!(n−r)!C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}
n
Total number of items
r
Number of items chosen

Worked example

Choose 3 from 10 (order does not matter)

Given
n = 10
r = 3
Calculation
C(10,3)=10!/(3!×7!)C(10,3) = 10! / (3! \times 7!)
Result
= 120 ways

Permutations P(n, r)

P(n,r)=n!(n−r)!P(n,r) = \frac{n!}{(n-r)!}
n
Total number of items
r
Number of items arranged (order matters)

Worked example

Arrange 3 from 10 (order matters)

Given
n = 10
r = 3
Calculation
P(10,3)=10!/7!P(10,3) = 10! / 7!
Result
= 720 arrangements

Frequently asked questions

What is the difference between combinations and permutations?

Combinations count selections where order does not matter: choosing 3 cards from 10 gives C(10,3) = 120. Permutations count arrangements where order matters: arranging 3 from 10 gives P(10,3) = 720. Always P(n,r) ≥ C(n,r).

What is the complement rule?

P(A') = 1 − P(A). The complement is the probability that the event does NOT occur. If there's a 30% chance of rain, there's a 70% chance of no rain.

What is the addition rule for probability?

P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events (cannot both occur), P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

How do I calculate the probability of multiple independent events?

For independent events, multiply their individual probabilities: P(A and B) = P(A) × P(B). For example, flipping heads twice: 0.5 × 0.5 = 0.25 (25%).

Key terms

Probability
A number between 0 and 1 representing the likelihood of an event. P = 0 means impossible; P = 1 means certain.
Factorial (n!)
The product of all positive integers up to n. 5! = 5×4×3×2×1 = 120.
Combination
An unordered selection of items from a set. C(n,r) = n! / (r! × (n-r)!).
Permutation
An ordered arrangement of items. P(n,r) = n! / (n-r)!.

Methodology, sources & limitations

This page shows the formula, assumptions, and limitations used for this estimate. Rates, thresholds, and other time-sensitive inputs can change, so verify them against the relevant official source before relying on a result.

For our review approach and the primary sources we consult, see the methodology and source policy. If you spot an outdated figure or formula, please contact us with a correction.