Conditional probability diagrams

Probability tree & bag

<p style="margin-bottom: 1.5rem">\(\mathrm{P}\left( A\cap B \right)=\mathrm{P}\left( A \right)\times \mathrm{P}\left( B \right)\)</p>
<p>\(\mathrm{P}\left( A|B \right)=\mathrm{P}\left( A \right)\)</p>

Events \(A\) and \(B\) are independent if the occurrence of either event does not affect the other. For nonzero conditioning probabilities, the following conditions are equivalent.

\[P(A\mid B)=P(A),\qquad P(B\mid A)=P(B),\qquad P(A\cap B)=P(A)P(B)\]

Independence tests

If \(A\) and \(B\) are independent, then \(A'\) and \(B'\) are independent and \(P(A'\cap B')=P(A')P(B')\). If \(P(A)\ne0\) and \(P(B)\ne0\), independent events are not mutually exclusive; conversely, mutually exclusive events are not independent.

Independent Events

Independent Events

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