Probability calculator

Choose a probability method from the question

Known assumptions or measured records? Distributions calculate consequences of supplied probabilities and parameters. Observed-count methods estimate proportions from records. Neither can manufacture a trustworthy real-world likelihood from an unknown input.

• Win bids, convert leads or see failures: binomial for independent opportunities with the same probability; Bernoulli for one. Select “at least” for reaching a target and “at most” for staying within capacity.
• Calls, arrivals or defects per fixed exposure: Poisson if arrivals are independent and the rate is stable. Use the same exposure in the historical rate and the question. Clusters and changing demand require another model.
• Waiting for successes: geometric or negative binomial count failures before the required successes. Total attempts = failures + required successes.
• Inspect a finite shipment: hypergeometric draws without replacement and requires the known marked count. Use a confidence interval instead when the population defect rate is unknown and you have sample records.
• Several outcome categories: multinomial for one exact category-count vector, with independent trials and category probabilities totaling one.
• Measured amounts or durations: choose normal, uniform, exponential, lognormal, gamma, beta, Weibull, triangular or logistic only when its shape and support fit. This workspace does not automatically fit or select a distribution.
• Statistical reference probabilities: central Student t, chi-square and F accept supplied degrees of freedom. They do not automatically perform a significance test or infer causation.
• Combine evidence: event relationships and conditional observed shares distinguish overlap from independence. Bayes updates a supplied prior using the signal rates in the correct direction.
• Estimate a success rate: enter observed successes and trials for a Wilson or exact Clopper–Pearson interval. A confidence interval is not a probability interval for next month’s count.

How to obtain the inputs

Use a defined cohort, interval or exposure from matching logs, inspections, comparable completed jobs or measured samples. Record the observation period and sample size. Exclude new/unmatched cases from conditional cohorts; account for bias, changing conditions and dependence. A historical rate is an estimate with uncertainty, not a permanent fact. Without representative records, label the numbers as assumptions and compare alternatives.

For lognormal inputs, compute the mean and standard deviation of the natural logarithms of positive observations. For gamma, scale is the reciprocal of rate. For normal, standard deviation is not a standard error. All bounds use the same units as the variable. A probability input of 0.25 means 25%; confidence level is explicitly entered as a percent.

Accuracy and honest limits

Methods are deterministic numerical implementations, checked against independent SciPy reference values, elementary examples, complement relationships and boundaries. This is not independent professional certification. Very small tails may underflow, near-one values can round, extreme percentiles or parameter combinations may be rejected, and sampled charts can omit far tails. The reported probability is computed from the full supported model, not from shaded pixels. No Monte Carlo simulation, automated distribution fitting, dependent-event model, medical diagnosis or guaranteed forecast is provided.

On-page calculations stay in your browser. Basic printing and result JSON downloads are free; results clear when inputs change. A bounded version is available through the API/MCP connection, with tighter parameter limits. For descriptive statistics and regression, use the Statistics Calculator.

What question or model fits your situation?

Choose by question, not by the result you hope to obtain. Distribution assumptions must match your data.

Which probability do you need?

At least includes the entered count. At most includes it too. Percentiles return the smallest count reaching the target.

Which continuous probability do you need?

Continuous single values have probability zero. Density height is not a probability.

Number of independent trials

Use matching independent trials, each with the same probability. Distribution/counting maximum 10,000; intervals allow up to 1,000,000 for Wilson.

Success probability per trial (decimal 0–1)

Use a justified historical rate from comparable independent opportunities, or an explicitly identified assumption. Do not enter 25 for 25%.

Required successes / items chosen

Negative binomial counts FAILURES before this many successes. Counting chooses this many distinct items without replacement.

Finite population size

Use the total number of items in the defined batch; maximum 1,000,000.

Known marked / defective items in population

A known fixed count, not an estimated defect percentage.

Expected events per interval / rate

Poisson: average event COUNT in the same exposure interval as the question. Exponential: events per one time unit, reciprocal of mean wait.

Mean / location

Normal: population mean in outcome units. Lognormal: mean of NATURAL LOG observations. Logistic: central location.

Standard deviation / scale

Normal: population SD. Lognormal: SD of natural logs. Gamma/Weibull/logistic: scale parameter, not automatically SD.

First shape parameter

A supplied or separately fitted shape in 0.05–10,000. Choosing a distribution does not fit it to observations.

Beta shape

Positive second beta shape, 0.05–10,000. Beta outcomes are between zero and one.

Minimum possible value

Lower finite bound, in the same units as threshold and upper bound.

Maximum possible value

Upper finite bound; must exceed minimum.

Most likely value (mode)

An assumed peak within the finite bounds. This does not prove the true outcome follows a triangle.

Degrees of freedom

Central distributions only; supply valid positive degrees of freedom, 0.05–10,000. Not inferred from a test.

Denominator degrees of freedom

Positive denominator degrees of freedom, 0.05–10,000.

Count / threshold / lower bound

Discrete: whole event count; geometric and negative binomial count failures, not total attempts. Continuous: outcome value in matching units.

Upper bound for a between query

Used only for a between query; at least the lower bound. Counts must be integers. Other queries ignore this field.

Cumulative percentile probability (decimal)

Used only for percentile queries. Supported range 0.0000000001–0.9999999999, excluding endpoints.

Relationship between events

Independent means occurrence of either event does not change the probability of the other.

P(A)

Marginal probability of event A, decimal 0–1.

P(B)

Marginal probability of event B, decimal 0–1.

P(A and B)

Actual or assumed overlap, at most the smaller marginal and at least max(0,P(A)+P(B)−1). Ignored if independence is selected.

Observed count: both A and B

Four non-overlapping counts from one matched observation cohort. Use explicit zero where none occurred.

Observed count: A but not B

Four non-overlapping counts from one matched observation cohort. Use explicit zero where none occurred.

Observed count: B but not A

Four non-overlapping counts from one matched observation cohort. Use explicit zero where none occurred.

Observed count: neither A nor B

Four non-overlapping counts from one matched observation cohort. Use explicit zero where none occurred.

Prior P(event)

Probability before seeing this signal, for the matching population. This must have a defensible source or be labeled an assumption.

P(signal given event)

Likelihood of the signal WHEN the event is true; this is not the probability of the event after a signal.

P(signal given no event)

Likelihood of the same signal when the event is false, for the same process/population.

Outcome/count, probability — one pair per line

2–20 rows. Multinomial: integer category count,decimal probability. Expected value: numerical outcome,decimal probability. Probabilities must total 1; commas separate the two fields, not thousands.

Observed successes

Count actual successes within the observed trials, including zero. Sample must be representative and independent for population inference.

Confidence level (%)

Choose 50–99.99. Repeated-sampling coverage, not a probability that a fixed population parameter is in this one interval.