The phrase complement of conditional probability refers to the probability that an event does not occur, given that some condition has occurred. If \(A\) is an event and \(B\) is the conditioning event with \(P(B) > 0\), the complement event is \(A^c\) (“not \(A\)”).
\[ P(A^c \mid B) = 1 - P(A \mid B) \quad \text{(valid whenever } P(B) > 0\text{).} \]
Meaning in words
Conditional probability \(P(A \mid B)\) is the probability of \(A\) restricted to the cases where \(B\) occurs. Therefore, \(P(A^c \mid B)\) is the probability that \(A\) does not occur among those same \(B\)-cases. Inside the conditioning set \(B\), either \(A\) happens or it does not—these two outcomes exhaust \(B\) and do not overlap.
Derivation from definitions
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Start with the definition of conditional probability:
\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0. \]
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Recognize that within \(B\), the events \(A\cap B\) and \(A^c \cap B\) partition \(B\):
\[ B = (A \cap B) \,\cup\, (A^c \cap B), \]
with \((A \cap B) \cap (A^c \cap B) = \varnothing\). -
Add probabilities of disjoint events:
\[ P(B) = P(A \cap B) + P(A^c \cap B). \]
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Divide both sides by \(P(B)\) (allowed because \(P(B) > 0\)):
\[ 1 = \frac{P(A \cap B)}{P(B)} + \frac{P(A^c \cap B)}{P(B)} = P(A \mid B) + P(A^c \mid B). \]
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Rearrange:
\[ P(A^c \mid B) = 1 - P(A \mid B). \]
Visualization: conditioning “shrinks” the sample space to \(B\)
Quick computation rule
Once \(P(A \mid B)\) is known, the complement is immediate:
\[ P(A^c \mid B) = 1 - P(A \mid B). \]
Worked numerical example
Suppose \(P(A \mid B) = 0.30\). Then:
\[ P(A^c \mid B) = 1 - 0.30 = 0.70. \]
Interpretation: among outcomes where \(B\) occurs, 70% correspond to “not \(A\)”.
Common confusions to avoid
| Expression | Meaning | Correct complement relationship |
|---|---|---|
| \(P(A^c \mid B)\) | Probability of not \(A\) given \(B\) | \(P(A^c \mid B)=1-P(A \mid B)\) |
| \(P(A \mid B^c)\) | Probability of \(A\) given not \(B\) | Not the complement of \(P(A \mid B)\) |
| \(P(A^c)\) | Unconditional probability of not \(A\) | \(P(A^c)=1-P(A)\) (different conditioning) |
Conditional probabilities such as \(P(A \mid B)\) and \(P(A^c \mid B)\) are defined only when \(P(B) > 0\).