Probability Calculator

Understanding probability for students and professionals

Probability quantifies uncertainty on a 0–1 scale. It underpins decision‑making, forecasting, quality control, risk analysis, and scientific inference. With clear inputs and assumptions, you can compute event probabilities, work with unions and conditionals, and use distributions to model repeated or random phenomena.

Core concepts

• Event probability: Favorable outcomes divided by total outcomes when results are equally likely.
• Union and intersection: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); intersection captures overlap.
• Conditional probability: P(A|B) = P(A ∩ B) / P(B) when P(B) > 0.
• Bayes theorem: Updates beliefs with evidence: P(A|B) = [P(B|A)·P(A)] / [P(B|A)·P(A) + P(B|¬A)·P(¬A)].

Common distributions

• Binomial: Number of successes in n independent trials with probability p. PMF: C(n,k)·p^k·(1−p)^(n−k). Mean = np; SD = √(np(1−p)).
• Poisson: Counts of events in a fixed interval at rate λ. PMF: e^(−λ)·λ^k / k!. Mean = λ; SD = √λ.
• Normal: Continuous bell‑shaped distribution with mean μ and SD σ. CDF gives probabilities up to x, often via z‑scores.

Best practices

• State assumptions: Independence, equal likelihood, and parameter sources (p, λ, μ, σ).
• Use consistent precision: Round final probabilities uniformly for clarity.
• Check domains: Probabilities belong in [0,1]; variances and counts are non‑negative.
• Validate inputs: Ensure k is within 0..n (binomial) and rates/SDs are positive.

Interpretation

A probability near 0 indicates rarity; near 1 indicates near certainty. Distribution results (PMF/CDF) describe single outcomes or cumulative chances. When evidence arrives (Bayes), update prior beliefs to posterior probabilities for better decisions.

Frequently asked questions

HOW TO calculate a simple event probability?

Steps: Choose Event probability, enter favorable and total outcomes, optionally select complement, then Calculate.

HOW TO compute the union of two events?

Select Union, provide P(A), P(B), and P(A ∩ B), choose P(A ∪ B), and calculate. For independence, P(A ∩ B) = P(A)·P(B).

HOW TO find conditional probability P(A|B)?

Choose Conditional, enter P(A ∩ B) and P(B), pick P(A|B) or P(B|A), then Calculate.

HOW TO use Bayes theorem?

Select Bayes, enter prior P(A), likelihoods P(B|A) and P(B|¬A), choose posterior P(A|B) or evidence P(B), and calculate.

HOW TO calculate binomial probabilities?

Choose Binomial, set n, p, and k, select PMF or CDF (≤ or ≥), then Calculate. Mean and SD are shown.

HOW TO calculate Poisson probabilities?

Select Poisson, set λ and k, choose PMF or CDF (≤ or ≥), then Calculate. Mean and SD are included.

HOW TO get normal distribution probabilities?

Choose Normal, enter μ and σ, provide x or interval a,b, select CDF, tail, or interval, and calculate. The tool returns z and probability.

HOW TO format and copy results?

Pick Output format = Plain, Compact, or CSV and use Copy result for reports and spreadsheets.

HOW TO embed this calculator on my website?

Copy the single‑file HTML page into your project, link it in navigation, or embed via iframe with a descriptive title.