Math calculator

Geometric Distribution Calculator

Find the chance the first success occurs on trial k. A checkable formula accompanies first-success probability instead of leaving an unexplained number.

Geometric Distribution inputs

Known quantities

Interpreting changes in Geometric Distribution

The geometric distribution requires k−1 failures followed by one success, giving (1−p)ᵏ⁻¹p under this trial-counting convention.

Read First-success probability against Trial of first success k, not in isolation. Use Success probability p to estimate the Geometric Distribution magnitude. When Success probability p is altered, label the new Geometric Distribution trial. Its First-success probability should not replace the original Geometric Distribution answer.

A numerical case for Geometric Distribution

With p=.25, first success on trial five has probability .75⁴·.25≈.0791.

When to reach for Geometric Distribution

It models waiting for a first sale, defect, response, win, or other Bernoulli success.

Some references count failures and begin at zero. This page counts the successful trial and begins k at one. If the Geometric Distribution assumptions do not fit, consider rth success.

What moves the Geometric Distribution result

How to label a Geometric Distribution result

Multiply k−1 failure probabilities and then p for success on trial k. Geometric Distribution also leads to fixed trials.

What can change the Geometric Distribution answer

Before carrying a Geometric Distribution result into later work, vary one of Trial of first success k, Success probability p and predict whether the answer should rise, fall, or remain unchanged. This sensitivity check is especially useful when similar fields represent totals, successes, intersections, or conditioning events.

For Geometric Distribution, a surprising response should trigger a model review before a rounding change. Confirm independence, replacement, event orientation, and row numbering in the source problem, because arithmetic cannot repair a sample space defined incorrectly.

A second verification of Geometric Distribution

Classify Trial of first success k before using Geometric Distribution. Confirm that Success probability p fits the allowable Geometric Distribution range.

Test a zero or certain-event Geometric Distribution boundary. Compare that simple Geometric Distribution case with Success probability p.

A hand-worked reference for Geometric Distribution need not duplicate the original numbers. Select an uncomplicated Trial of first success k, retain Success probability p, and estimate First-success probability. The reference gives the original Geometric Distribution answer a meaningful scale check.

Questions about Geometric Distribution

What is the smallest k?

One.

What is the mean?

1/p trials.

Why k−1 failures?

Trial k is the success.

Does the model have memory?

No.