Probability Calculator
Compute single-event and combined probabilities, plus binomial outcomes.
Single, combined & binomialCompute basic probabilities from favourable outcomes, combine independent or mutually exclusive events, and evaluate binomial probabilities – all in one tool, with clear formulas and explanations.
Good for: homework checks, teaching, probability intuition, quick sanity checks before sending a report or implementing logic in code or spreadsheets.
Single event probability – favourable vs total outcomes
Use this mode when all outcomes are equally likely. Example: probability of drawing a red ball from an urn, rolling an even number on a fair die, or picking a specific card from a shuffled deck.
Outcomes that count as “success”.
All equally likely outcomes.
Core definitions: events, outcomes and probability
In probability theory, we start from the sample space (all possible outcomes) and define an event as any collection of outcomes we care about. If all outcomes are equally likely, the probability of an event A is
P(A) = number of favourable outcomes / number of possible outcomes.
Example: rolling a 4 on a fair die: P(4) = 1/6; rolling an even number: P({2,4,6}) = 3/6 = 1/2.
Independent vs mutually exclusive events
The way we combine probabilities depends on how events are related:
- Independent events do not affect each other. For independent A and B: P(A and B) = P(A) · P(B).
- Mutually exclusive events cannot occur at the same time. For mutually exclusive A and B: P(A and B) = 0 and P(A or B) = P(A) + P(B).
- In general, P(A or B) = P(A) + P(B) − P(A and B).
Binomial distribution in practice
The binomial model is a workhorse in statistics, quality control and A/B testing. It applies when:
- You repeat the same trial a fixed number of times n,
- each trial has just two outcomes (success/failure),
- the probability of success p is constant, and
- trials are independent.
Under these assumptions, the random variable X = number of successes follows a binomial distribution. The calculator uses this distribution to compute probabilities like:
- “Exactly k successes” P(X = k),
- “At most k successes” P(X ≤ k),
- “At least k successes” P(X ≥ k).
FAQ: using this probability calculator correctly
Can I mix fractions, decimals and percentages?
In the combined events section you can enter probabilities as fractions (3/10), decimals (0.3) or percentages (30) – the calculator normalises them internally as decimal probabilities between 0 and 1. For the binomial section, use a decimal p between 0 and 1.
What if my trials are not independent?
If trials influence each other (for example, drawing cards without replacement), the binomial model is only an approximation. For small samples this can make a noticeable difference, so prefer exact combinatorial calculations or specialised models (such as the hypergeometric distribution).
Can I use this for real-world risk assessment?
You can use this calculator to clarify assumptions and do order-of-magnitude checks. However, real risk assessment often requires more complex models, empirical data and domain expertise. For high-impact decisions, always review your reasoning with a statistician or subject-matter expert.
Formula (LaTeX) + variables + units
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Formula used For X ~ Bin(n, p), the probability of exactly k successes is P(X = k) = C(n, k) · pk · (1 − p)n − k where C(n, k) is the binomial coefficient “n choose k”.
P(A) = number of favourable outcomes / number of possible outcomes. Example: rolling a 4 on a fair die: P(4) = 1/6; rolling an even number: P({2,4,6}) = 3/6 = 1/2.
- No variables provided in audit spec.
- NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures - FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.