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I have come across both $P(\dots)$ and $\Pr(\dots)$ being used to represent probabilities. Is there any difference in the meaning of these notations, or are they just different shorthands?

I seem to come by $\Pr(\dots)$ more often in Bayesian probability contexts, though I wouldn't say that's a rule.

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  • $\begingroup$ I use both indifferently, but if you have a whole bunch of them in an expression, using $P$ instead of $\Pr$ may make the difference between whether or not the expression fits on a single line. $\endgroup$
    – MPW
    Commented Jul 22, 2014 at 23:36
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    $\begingroup$ As you might know, the average of a certain variable $x$ can be expressed as $\langle x \rangle$ or $\bar{x}$ and probably there are other ways as well. No notation is fundamentally preferred, people learn a certain way, stick to it and eventually if they publish a book you get to read the notation they like. $\endgroup$
    – Squirtle
    Commented Jul 22, 2014 at 23:48
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    $\begingroup$ I personally use $\mathbb P$ (blackboard bold). $\endgroup$
    – Math1000
    Commented Feb 3, 2020 at 19:08

3 Answers 3

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They are just different conventions.   They don't signify any different meaning.

I personally find the $\Pr$ notation most useful when the discussion involves combinatorics.   It distinguishes probability somewhat from permutation. (Unless you use ${^n{\rm P}_r}$ ...)

It also has that convenient LaTeX command \Pr which renders it in times roman font, and with some space padding, which helps it stand out in a line of multiplied probabilities using just a few keystrokes.

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They are just different notation. Some authors even use the blackboard bold font: $\mathbb{P}$. What matters is what's inside of the subsequent parentheses (or sometimes brackets, [].)

Several notation species exist for expectation ($E, \text{E},\mathbb{E}$) and variance ($V, \text{V},Var, \mathbb{V}$) too, but they all have the same definition.

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In probability theory, a typical set-up involves just one probability space $(\Omega, \mathrm P)$ and many random variables $X$, $Y$. Each of them has a distribution $P_X$, $P_Y$ defined in terms of the underlying probability measure $\mathrm P$, for example $P_X(A) = \mathrm P(X \in A)$. In this context, $\mathrm P(E)$ is the probability measure which is anchored to the real world notion of "probability" and events $E$. For example, if $X$ is a random variable modeling a 6-sided die, then $\mathrm P(X = 6) = 1/6$ corresponds really to "the probability of the event that my die is showing a 6 is 1/6". The notation $\operatorname{Pr}$ typically refers to that real-world probability.

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