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Complement Rule

P(Ac)=1−P(A)P(A^{c}) = 1 - P(A)

Worked example: P(A) = 0.25 → complement 0.75 — press Try an example to run it live, then adjust anything.

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Complement Rule explained

P(A)P(Aᶜ)

Every trial ends either in A or in not-A, and those two possibilities carry all the probability there is — so they must add to 1. Probabilities on this site are decimals from 0 to 1 (the % unit is there if you prefer percentages), and the complement rule is the cheapest trick in the subject: if a weather model gives a 0.3 chance of rain, the chance of a dry day is 1 − 0.3 = 0.7, no further modelling required.

Its real power is turning hard problems inside out. "At least one" questions are almost always easier as "not none": the probability of at least one head in five coin tosses means adding five separate cases, while the complement is a single product, 1 − 0.5⁵ = 1 − 0.03125 = 0.96875. Pascal and Fermat leaned on exactly this inversion in their 1654 letters. The classic error is subtracting from the wrong whole — the complement of "at least two" is "zero or one", not "at most two".

Complement Rule formula

P(Ac)=1−P(A)P(A^{c}) = 1 - P(A)
Where
  • P(Ac)P(A^{c})= Probability the event does NOT occur
  • P(A)P(A)= Probability the event occurs

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