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What is the rule for calculating the probability of an event?
in Data Science & Statistics by Diamond (58,497 points) | 28 views

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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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