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What is the rule for calculating the probability of an event?
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If $A$ is an event in the finite sample space $\mathcal{S}$, then $P(A)$ equals the sum of the probabilities of the individual outcomes comprising $A$.

Proof:

To prove this, let $E_{1}, E_{2}, \ldots, E_{n}$ be the $n$ outcomes comprising event $A$, so that we can write $A=E_{1} \cup E_{2} \cup \cdots \cup E_{n}$. Since the $E$ 's are individual outcomes, they are mutually exclusive, and by Theorem $3.4$ we have
\begin{aligned} P(A) &=P\left(E_{1} \cup E_{2} \cup \cdots \cup E_{n}\right) \\ &=P\left(E_{1}\right)+P\left(E_{2}\right)+\cdots+P\left(E_{n}\right) \end{aligned}
which completes the proof.

by Diamond (58,497 points)

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