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I'm taking an introductory statistics course for engineers, so my textbook can be a little loose when it comes to formal definitions, and I have to look certain things up myself.

I know that the probability function is defined as $P: \mathcal{P}(S) \rightarrow [0,1]$, where $\mathcal{P}(S)$ is the powerset of the sample space. However, I sometimes see others (namely my professor) write $P(e)$ where $e \in S$. Are they being sloppy? Did they mean $P(\{e\})$?

MJD
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user168651
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    Yes, they are most likely just being sloppy about the notation. Like you're saying, $P(e)$ doesn't really make sense. – Snildt Jan 24 '22 at 16:10
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    It isn't too uncommon to lessen the notation regarding probabilities though. When you consider a random variable $X$ you might write $P(X\in A)$, but "$X\in A$" isn't a set in the powerset either, it's just shorthand for ${\omega \in \Omega : X(\omega) \in A}$ – Snildt Jan 24 '22 at 16:18
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    This abuse of notation (writing $P(e)$) is in fact much more common than writing the rigorously-correct $P({E}).$ remark #6 – ryang Jan 24 '22 at 16:27

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