Find the chance the first success occurs on trial k. A checkable formula accompanies first-success probability instead of leaving an unexplained number.
The geometric distribution requires k−1 failures followed by one success, giving (1−p)ᵏ⁻¹p under this trial-counting convention.
This definition sets the boundary between geometric distribution and a neighboring calculation with similar inputs.
Multiply k−1 failure probabilities and then p for success on trial k. Geometric Distribution also leads to fixed trials.
With p=.25, first success on trial five has probability .75⁴·.25≈.0791.
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.
Changing only one field helps distinguish a data-entry problem from the intended behavior of geometric distribution.
A field name is part of the formula. Match the problem's quantities to trial of first success k and success probability p, then check that they share the scale assumed by the geometric distribution relationship.
Larger k adds another failure factor, so exact probabilities decay geometrically for 0<p<1. This is a stronger check than judging the answer only by how many decimal places it shows.
Match the reported precision to trial of first success k and success probability p, not to the number of digits the browser can display. Preserve an exact form when it communicates the geometric distribution structure more clearly than a decimal.
Negative binomial generalizes the waiting question to the rth success. The formula panel makes the chosen definition explicit.
The minimum audit trail is short: trial of first success k and success probability p, their units, and the formula beside the answer. It is enough to distinguish this calculation from a similar-looking shortcut.
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.
One.
1/p trials.
Trial k is the success.
No.