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Q&A

"A occurred" vs. "something must happen"

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  1. Why doesn't "Something must happen" mean $s_{actual} \in A$?

  2. Scilicet, doesn't "A occurs" mean the same thing as "something must happen"? Something must happen. $\iff$ Some event must happen. $\iff$ At least one event must happen $\iff$ Call this event A. Then A occurred.

Blitzstein, Hwang. Introduction to Probability (2019 2 ed). p 6.

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As it currently stands, your first question is inconsistent with what is in your image. (2 comments)

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$A$ is a specific event, specified beforehand.

Let's have an example. Let a random experiment be rolling a six-sided die and looking at what face lands up. The sample space is $S=\{1,2,3,4,5,6\}$.

What is event $A$? Let's specify it before the experiment is done. Let $A$ be the event that an even number lands up, that is, $A=\{2,4,6\}$.

Say the experiment is done and the outcome is $2$. Did $A$ occur? Since $2\in A$, the answer is yes.

Say the experiment is done again and there was an outcome $s$, but we don't know what it was. Can we say $s\in A$? No. Can we say $s\not\in A$? No. But we can say that something happened, that is, $s\in S$.

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