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

The **intersection** \(A\cap B\) is the event that both \(A\) and \(B\) occur. The **union** \(A\cup B\) is the event that at least one of \(A\) or \(B\) occurs.

Two-event union

\[P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(B\cap C)-P(A\cap C)+P(A\cap B\cap C)\]

Three-event union

\[(A\cup B)'=A'\cap B',\qquad (A\cap B)'=A'\cup B'\]\[P((A\cup B)')=1-P(A\cup B)=P(A'\cap B')\]\[P((A\cap B)')=1-P(A\cap B)=P(A'\cup B')\]

Set and probability forms

If \(P(A)=0.5\), \(P(B)=0.4\) and \(P(A\cup B)=0.7\), then \(P(A\cap B)=0.5+0.4-0.7=0.2\) and \(P(A'\cap B')=1-0.7=0.3\).

**Exam check:** every region of the Venn diagram must be non-negative and the regions must add to \(1\); here \(0.3+0.2+0.2+0.3=1\).

Set Operations and Probability Formulae

Set Operations and Probability Formulae

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