Alternative formulation of complexity class $BPP$ In Aurora and Barak, they give the following alternative definition of $BPP$:



What is the meaning of the subscript to $Pr$? Is that $Pr_{r \in_R \{0,1\}^{p(|x|)}}$? My guess is this is supposed to represent $poly(|x|)$ many coin toss results, but what is $R$? And why is this a subscript to $Pr$?
The subscript to $\Pr$ denotes which random variable the probability is over. In this case the probability is over $r$, which is samples uniformly at random (this is what is conveyed by $\in_R$) from $\{0,1\}^{p(n)}$. You can think of $r$ as representing a sequence of $p(n)$ many random coin tosses.