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It is known that in statistics $P(X)$ represents the probability of $X$. My question: is it WRONG to use the letter $P$ to represent an event, where $P(P)$ represents the probability of $P$.

For example, let $P$ represent the event of having pizza for dinner. Then, $P(P)$ represents the probability of having pizza for dinner.

In summary, is the letter $P$ reserved or not such that we can use it to represent general events?

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    $\begingroup$ In mathematics, you can use any notation you like as long as you clearly define the symbols you use and the resulting notation is unambiguous to a reader. It will generally be easier for both you and the reader if you fit in with common usage, but this is just convenience and following common usage is not the same as symbols being "reserved". By the way, people use many notations for the probability of an event other than P(X), .e.g., Pr(X) is also very common and there are lots of other variants. $\endgroup$ – Gordon Smyth Jun 6 '20 at 4:43
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    $\begingroup$ @: do you want to post your comment(s) as an answer? Better to have a short answer than no answer at all. Anyone who has a better answer can post it. I would add that while $P(P)$ is unambiguous, it is needlessly confusing. Yes, you can give your dog the name "Cat", but if you post to Pets.SE about a problem Cat has, don't be surprised if you get answers that are not applicable to dogs. $\endgroup$ – Stephan Kolassa Jun 6 '20 at 6:39
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    $\begingroup$ I think that last part would be an indication that the potential ambiguity was a real ambiguity (in the sense that people would be misled by it); I think that Gordon covered that with "unambiguous to a reader". $\endgroup$ – Glen_b Jun 6 '20 at 7:38
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    $\begingroup$ @GordonSmyth: thanks, upvoted! Per that meta thread ("don't let the perfect be the enemy of the good"), better to have a short answer than none at all. If it's not enough for the OP, they can always post a new question to point out where the answer was not enough, or post a bounty. $\endgroup$ – Stephan Kolassa Jun 6 '20 at 12:27
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    $\begingroup$ Related: Which notation and why: P() , Pr() , Prob() , or ℙ() $\endgroup$ – Alexis Jun 6 '20 at 13:57
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In mathematics, you can use any notation you like as long as you clearly define the symbols you use and the resulting notation is unambiguous to a reader. Having said that, it will generally be easier for both you and the reader if you fit in with common usage, but this is just for convenience and following common usage is not the same as symbols being "reserved".

People use many notations for the probability of an event $X$ other than $P(X)$, for example Pr$(X)$ is also very common and there are lots of other variants such as $\pi(X)$ or $p(X)$.

I would strongly discourage the use of $P(P)$. While that notation is technically possible, because it is possible to keep the probability function $P()$ separate from the event $P$, it nevertheless is needlessly confusing. If you do want to use $P$ to represent an event, then it would be wise to prepare for that by choosing a different notation other than $P()$ for the probability function.

Mathematical notation, like language in general, is a fluid thing. The same author may use different notation for the same concept at different times, depending on the audience and the context and on what other notation needs to be used in the same document. It's not necessary to do exactly what other people do, but it is necessary to be clear and unambiguous.

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  • $\begingroup$ While it can be done, notation overloading is a recipe for making mistakes in the long run. $\endgroup$ – pjs Jun 6 '20 at 19:37
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    $\begingroup$ @Alexis Thanks for the link, which I've added to my answer. $\endgroup$ – Gordon Smyth Jun 6 '20 at 23:51

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