Probability theory MOC

Conditional probability

Conditional probability allows the investigation of how the knowledge of one event occurring effects the knowledge of the other one. Given a probability model , and two events , the conditional probability of given is prob

unless , in which case .¹ The function forms a probability measure on the same space as .

Properties

\mathbb{P}(A\cap B)= \mathbb{P}(B)\mathbb{P}(A\mid B) = \mathbb{P}(A)\mathbb{P}(B \mid A)

\frac{\mathbb{P}(A\mid B)}{\mathbb{P}(A^c \mid B)} = \frac{\mathbb{P}(B \mid A)}{\mathbb{P}(B \mid A^c)} \frac{\mathbb{P}(A)}{\mathbb{P}(A^c)}

\begin{align*} \mathbb{P}(B)=\sum_{i=1}^n\mathbb{P}(B\mid A_{i})\mathbb{P}(A_{i}) \end{align*}

See also

• Conditional distribution
• Conditional independence
• Conditional expected value
• Conditional variance

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Footnotes

1. Since this may be considered an impossible scenario. ↩