Probability Calculator
Calculate the probability of single and multiple independent events.
Calculate probabilities for single events, two combined events, and repeated trials.
Probability Distribution (Binomial)
| Successes (k) | P(X = k) | P(X ≤ k) | P(X ≥ k) |
|---|
</> Embed this calculator
Free to embed on any website — no sign-up needed. Paste this snippet where you want the calculator to appear.
<div class="yootil-widget" data-tool="probability-calculator"></div>
<script src="https://yootils.com/widgets/v7/widgets.js" integrity="sha384-+/J+UJUQkd8qBlJ2y9qW3wNk89iZUobwuVdo6dqpCvoanps684ZTa7cGCJ5L1AQc" crossorigin="anonymous" async></script>
Understanding Probability: Key Concepts and Terms
Basic probability
P(A) = (favourable outcomes) ÷ (total possible outcomes). It always falls between 0 (impossible) and 1 (certain). The complement P(A') = 1 − P(A) is the probability the event does NOT occur. To express a probability as a percentage, use the percent calculator.
Independent vs mutually exclusive
Independent events don't affect each other — P(A∩B) = P(A)×P(B). Mutually exclusive events can't both occur simultaneously — P(A∩B) = 0. Note: independent and mutually exclusive are NOT the same thing. Probabilities are often expressed as fractions — the fractions calculator can help simplify them.
Conditional probability
P(A|B) = P(A∩B) ÷ P(B) is the probability of A given that B has already occurred. This is the basis of Bayes' theorem: P(A|B) = P(B|A)×P(A) ÷ P(B).
Binomial distribution
P(X = k) = C(n,k) × p^k × (1−p)^(n−k) gives the exact probability of k successes in n independent trials each with success probability p. The expected value is n×p and standard deviation is √(n×p×(1−p)).
Worked example
Rolling a fair six-sided die: P(rolling a 4) = 1/6 ≈ 16.7%; P(not rolling a 4) = 5/6 ≈ 83.3%. Rolling at least one 6 in three rolls: P = 1 − (5/6)³ = 1 − 0.5787 = 42.1%. Binomial: flipping 10 fair coins, P(exactly 7 heads) = C(10,7) × 0.5^7 × 0.5^3 = 120 × 0.0078 ≈ 11.7%.
Related calculations
Odds are usually quoted as percentages, so the percentage calculator converts between the two, and the fractions calculator handles the fractional form used in betting odds.
Evidence & Methodology
How This Page Is Grounded
Method
Uses complement, addition and multiplication rules for single events, independent events and repeated Bernoulli trials.
Important limitation: Combined-event answers assume independence where stated; real-world events may be dependent.
Primary Sources
- Probability OpenStax
Quality Checks
Probabilities of 0 and 1, complements and repeated-event examples are checked.
See how sources are selected and corrections are handled in our Editorial & Calculation Methodology, and which automated checks this page has to pass in How We Test.
Frequently Asked Questions
How is basic probability calculated?
Probability is calculated as P(A) = favourable outcomes ÷ total possible outcomes, and always falls between 0 (impossible) and 1 (certain). The complement — the probability an event does NOT occur — is P(A') = 1 − P(A). Enter any value in the probability calculator above to explore these relationships instantly.
What is the difference between independent and mutually exclusive events?
Independent events don't affect each other — P(A∩B) = P(A)×P(B). Mutually exclusive events cannot both occur simultaneously — P(A∩B) = 0. These are not the same thing: independent events can both happen, mutually exclusive events cannot. The probability calculator handles both cases separately.
What is the binomial distribution and when does it apply?
The binomial distribution gives the probability of exactly k successes in n independent trials each with success probability p, using P(X=k) = C(n,k) × p^k × (1−p)^(n−k). Use the binomial section of the probability calculator to find the chance of any outcome across repeated trials.