Which notation and why: $\text{P}()$, $\Pr()$, $\text{Prob}()$, or $\mathbb{P}()$ Are these merely stylistic conventions (whether italicized or non-italicized), or are there substantive differences in the meanings of these notations?
Are there other notations meaning "the probability of" that should be considered in this question?
 A: I've seen all three used in different undergrad classes and as far as I know, they're stylistic differences and all represent probability as you're thinking of it.
One other notation I've seen is in Sheldon Ross's "Introduction to Probability Theory", where $\mathbf{P}$ represents a probability matrix.  He also uses $\pi_i$ as a notation for limiting probability, which a sequence of probabilities $(p_i)$ converges to.
A: Stylistic conventions, mainly, but with some underlying rationale.
$\mathbb{P}()$ and $\Pr()$ can be seen as two ways to "free up" the letter $\text{P}$ for other use—it is used to denote other things than "probability", for example in research with complicated and extensive notation where one starts to exhaust available letters.
$\mathbb{P}()$ requires special fonts, which is a disadvantage. $\Pr()$ may be useful when the author would want the reader to think of probability in abstract and general terms, using the second lower-capital letter "$r$" to disassociate the symbol as a whole from the usual way we write up functions.
For example, some problems are solved when one remembers that the cumulative distribution function of a random variable can be written and treated as a probability of an "inequality-event", and apply the basic probability rules rather than functional analysis.
In some cases, one may also see $\text {Prob}()$, again, usually in the beginning of an argument that will end up in a specific formulation of how this probability is functionally determined.
The italics version $P()$ is also used, and also in lower-case form, $p()$—this last version is especially used when discussing discrete random variables (where the probability mass function is a probability).
$\pi(\;,\;)$ is used for conditional ("transition") probabilities in Markov Theory.
A: This makes me think of Meyn and Tweedie's book. They use $P$ to denote the transition kernel for a Markov chain, and $\mathsf{P}$ for the law of the entire chain on $\mathsf{X}^{\infty}$. This answer is specific to Markov chains, but the distinction is obviously important.
The difference between $P$ and $\mathbb{P}$ (and $E$ and $\mathbb{E}$) from book to book, is just for aesthetic appeal, in my opinion. I can't generalize where I see $\Pr()$ or $\text{Prob}()$, really.
