# 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.