Complement rule of probability

The complement rule of probability states P(not A) = 1 − P(A), the chance an event fails to happen; it is the standard shortcut for 'at least one' problems.

ByHoang TruongUpdated July 11, 2026

See it move

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A shipping company finds that 15% of parcels arrive later than promised, so P(late) is 0.15. Because every parcel either arrives late or arrives on time, never both and never neither, the two probabilities must sum to exactly 1: P(on time) is 1 minus 0.15, or 0.85, without needing to calculate it directly.

Where it fits

TopicProbability & DistributionsCoreSubjectData Analysis & StatisticsCore

The formula

LaTeX

P(A^c) = 1 - P(A)

Variables

$A$: The event of interest
$P (A)$: Probability that event A occurs
$P (A^{c})$: Probability that event A does not occur

Gives the probability that an event does not occur, from the probability that it does.

LaTeX

P(\text{at least one}) = 1 - (1-p)^n

Variables

$p$: Probability the event occurs on a single trial
$n$: Number of independent trials

Gives the probability that an event happens at least once across n independent, identical trials.

Check yourself

PracticeCORE

A machine has 4 independent sensors, each with a 0.05 probability of malfunctioning. What is the probability that at least one sensor malfunctions?

Select an answer to check your understanding.