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Plain TeX defines \Pr as

\def\Pr{\mathop{\rm Pr}}

LaTeX and ConTeXt define it similarly (with some syntax sugar around the choice of the font). To me, a more reasonable definition is

\def\Pr{\mathop{\rm Pr}\nolimits}

so that constructs like \Pr^{f}(X) look nice.

Are there any scenarios where the tradition TeX definition is more appropriate than the latter?

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I can't speak to the design decisions made in plain TeX, LaTeX, ConTeXt, or amsmath (which reimplements \Pr). My guess is the reason it doesn't have \nolimits is because it is common to see \Pr_x denoting the probability over the choice of x.

I agree that \Pr^f(X) looks quite strange with the normal definitions.

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To me even \Pr_X looks better with \nolimits. Are their branches of math/comp sci where authors use \Pr_X with the default definition of \Pr? – Aditya Sep 30 '10 at 16:57
@Aditya: I'm not a mathematician (I just play one on the internet). I'm positive that I've seen it in computer science papers. But then, those are almost universally written in LaTeX so it seems reasonable to assume that people don't change the definitions. The concentration of measure book sitting on my desk has the \nolimits version. The information theory book on my desk doesn't seem to have any subscripts for \Pr. It seems like it's becoming more common to use \mathbb P instead of \Pr. I've never seen \mathop{\mathbb P}\limits_X though. – TH. Sep 30 '10 at 20:26
Thanks TH. I also see \mathbb P or mathds P (from dsfont package) more often, and always with \nolimits. – Aditya Oct 1 '10 at 2:02

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