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Probability Calculator

Calculate basic probability, combinations (nCr), permutations (nPr), and complement probability. Instant results with formulas.

Basic Probability P = Favorable / Total

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How to Use This Tool

  1. Choose a mode: Basic Probability, Combinations, Permutations, or Multiple Events.
  2. For Basic Probability, enter the number of favorable outcomes and the total number of outcomes.
  3. For Combinations or Permutations, enter n (total items) and r (items chosen or arranged).
  4. For Multiple Events, enter the probability (between 0 and 1) for each independent event, adding or removing events as needed.
  5. Review the result along with the percentage, complement, or detailed explanation shown below the inputs.

Formula & How It Works

Basic Probability

P(event) = Favorable Outcomes / Total Outcomes

The chance an event happens, expressed as a number between 0 (impossible) and 1 (certain).

Complement Rule

P(not A) = 1 − P(A)

The probability that an event does NOT happen is one minus the probability that it does.

Combinations

C(n, r) = n! / (r! × (n − r)!)

Counts how many ways to choose r items from n items when the order does not matter.

Permutations

P(n, r) = n! / (n − r)!

Counts how many ordered arrangements of r items can be made from n items.

Practical Examples & Common Use Cases

Example: Rolling a 4 on a Die

Favorable outcomes = 1, total outcomes = 6, so P = 1 / 6 ≈ 0.1667, or 16.67%. The complement (not rolling a 4) is 1 − 0.1667 = 0.8333, or 83.33%.

Example: C(10, 3) and P(10, 3)

Choosing 3 items from 10 without regard to order: C(10,3) = 10! / (3! × 7!) = 120 ways. Arranging 3 of those 10 items in order: P(10,3) = 10! / 7! = 720 ways.

Example: Two Independent Events

If P(A) = 0.5 and P(B) = 0.3, then P(A and B) = 0.5 × 0.3 = 0.15. P(at least one) = 1 − (1 − 0.5) × (1 − 0.3) = 1 − 0.35 = 0.65.

Frequently Asked Questions

Probability measures the likelihood of an event. It ranges from 0 (impossible) to 1 (certain). P = favorable outcomes / total outcomes.

Combinations count selections where order does not matter (C(5,2) = 10 ways to pick 2 from 5). Permutations count arrangements where order matters (P(5,2) = 20 ordered pairs).

The complement of event A is "A does not occur". P(not A) = 1 − P(A). Example: if P(rain) = 0.3, then P(no rain) = 0.7.

Multiply the individual probabilities together. P(A and B) = P(A) × P(B). For "at least one", use P = 1 − P(none occur) = 1 − (1−P(A)) × (1−P(B)).

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