Geometric distribution calculator

Values of p outside zero to one are rejected with invalid_input so homework answers and production retries stay honest.

What the geometric model covers and how probability is defined

If your textbook writes P of Y equals k as q to the k times p for k starting at zero, translate by setting this tool’s k to Y plus one, or compare only the shared p and the PMF formula printed in the JSON.

How to call the API and read the JSON response

Two formula strings document the convention so downstream UIs can show the same definition humans searched for. Classic check: p equals zero point two five and k equals three yields probability zero point one four zero six two five, expected trials four, and variance twelve. Another check: p equals one and k equals one yields probability one and expected trials one; the same p with k equals two yields probability zero. Send a JSON body with two required fields. Field p is the success probability on each trial; aliases probability, success_p, and prob are accepted. It must be a finite number strictly greater than zero and at most one. Field k is the trial index of the first success you care about; aliases trial, n, and trials map to the same meaning. k must be a whole integer from one through one million. A successful response echoes the normalized p and k, the failure probability q, and the PMF as both a trimmed decimal string (probability) and an IEEE double (probability_number) plus a percentage string. It also returns expected_trials as one over p, variance as q over p squared, and the CDF at k as cdf and cdf_number. Pricing is a flat $0.002 per successful request with unit item and zero marginal unit price. Prefer probability (string) when you need stable display digits; use probability_number when you will chain further floating-point math in a client script.

Practical tips, limits, and how this differs from related tools

Do not confuse this landing with binomial tools or combination counters elsewhere in the kit: the binomial law counts how many successes occur in a fixed number of trials, while the geometric law stops at the first success and treats the trial index as random. Dice-sum probability landings answer a different search intent—sums of polyhedral faces—not repeated Bernoulli waits. If you need the failures-before-success parameterization, convert: number of failures before first success is X minus one, with expectation q over p; this calculator keeps the trial-number form because that matches the phrase “first success on trial k” in search queries. Stay inside k at most one million so log-space evaluation of the PMF stays cheap in both Worker and browser. For classroom demos, prefer small textbook pairs—p equals one half with k equals one through five, or p equals one tenth with k equals ten—so students can expand (one minus p) to the power (k minus one) by hand and match the API line by line. When modeling software retries, map “attempt number” to k and the observed success rate to p; expected_trials then tells you the mean number of attempts before a healthy endpoint answers, and the CDF tells you how often you finish within a budgeted attempt limit. Failed validation never spends prepaid balance, so exploratory scripts can probe edges cheaply. Prefer the printed formula fields when documenting homework solutions, and prefer expected_trials over ad-hoc spreadsheet inverses of p so every teammate shares the same definition of the mean wait.