Editing Probability (section)

===Not (necessarily) mutually exclusive events===

If the events are not (necessarily) mutually exclusive then<math display="block">P\left(A \hbox{ or } B\right) = P(A \cup B) = P\left(A\right)+P\left(B\right)-P\left(A \mbox{ and } B\right).</math> Rewritten,<math display="block">
P\left( A\cup B\right)  =P\left( A\right)  +P\left( B\right)  -P\left( A\cap B\right)  
</math>

For example, when drawing a card from a deck of cards, the chance of getting a heart or a face card (J, Q, K) (or both) is <math>\tfrac{13}{52} + \tfrac{12}{52} - \tfrac{3}{52} = \tfrac{11}{26},</math> since among the 52 cards of a deck, 13 are hearts, 12 are face cards, and 3 are both: here the possibilities included in the "3 that are both" are included in each of the "13 hearts" and the "12 face cards", but should only be counted once.

This can be expanded further for multiple not (necessarily) mutually exclusive events. For three events, this proceeds as follows:<math display="block"> 
\begin{aligned}P\left( A\cup B\cup C\right)  =&P\left( \left( A\cup B\right)  \cup C\right)  \\ =&P\left( A\cup B\right)  +P\left( C\right)  -P\left( \left( A\cup B\right)  \cap C\right)  \\ =&P\left( A\right)  +P\left( B\right)  -P\left( A\cap B\right)  +P\left( C\right)  -P\left( \left( A\cap C\right)  \cup \left( B\cap C\right)  \right)  \\ =&P\left( A\right)  +P\left( B\right)  +P\left( C\right)  -P\left( A\cap B\right)  -\left( P\left( A\cap C\right)  +P\left( B\cap C\right)  -P\left( \left( A\cap C\right)  \cap \left( B\cap C\right)  \right)  \right)  \\ P\left( A\cup B\cup C\right)  =&P\left( A\right)  +P\left( B\right)  +P\left( C\right)  -P\left( A\cap B\right)  -P\left( A\cap C\right)  -P\left( B\cap C\right)  +P\left( A\cap B\cap C\right) \end{aligned} 
</math>It can be seen, then, that this pattern can be repeated for any number of events.