Find one or more successes across identical independent trials. Input changes update both at-least-one probability and the supporting steps.
At least one success is the complement of all trials failing, so its probability is 1−(1−p)ⁿ. Probability of At Least One can also be compared with complement rule.
The operation is a compact expression of the probability of at least one definition above.
Raise the single-trial failure probability to n, then subtract that all-fail probability from one.
With p=0.2 over eight trials, 1−0.8⁸≈0.8322.
The shortcut handles repeated attempts, component failures, exposure chances, and sampling with replacement. A related application of Probability of At Least One is unequal probabilities.
Trials must be independent and use the same p. Changing probabilities or drawing without replacement needs a different model.
When probability of at least one looks surprising, restore the sample and vary success probability per trial by itself.
For this probability of at least one calculation, the labels success probability per trial and independent trial count carry mathematical meaning. A transposed entry can remain numerically valid while describing an entirely different setup.
Adding trials cannot lower the result when p is fixed; for positive p it approaches one. A small controlled input change is enough to test the expected direction.
An exact expression can preserve factors, radicals, or π that a decimal hides. Use the representation suited to the next task and label it clearly as at-least-one probability.
The independent-events page accepts unequal probabilities, while this page repeats the same p. Checking the requested noun is often enough to select the right model.
Reproducibility here depends on the inputs more than the interface. Preserve success probability per trial and independent trial count, the operation shown, and enough unrounded digits for the next calculation.
Compare Probability of At Least One with a neighboring model using the same Success probability per trial, Independent trial count only after stating the changed assumption. For example, replacement separates binomial from hypergeometric sampling, while order separates a repeated permutation from a repeated combination.
For Probability of At Least One, the two answers need not be close, but their relationship should be explainable. Preserve the setup and a concise reason for choosing this model so a later reader does not reuse the number after silently changing the experiment.
It can exceed one and double-count outcomes.
The result is zero.
The result is zero.
No.