# Is there any difference between $P$ and $\Pr$ to represent probabilities?

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.

• 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.
– MPW
Jul 22, 2014 at 23:36
• 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. Jul 22, 2014 at 23:48
• I personally use $\mathbb P$ (blackboard bold). Feb 3, 2020 at 19:08

## 3 Answers

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.

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.

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.