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In Aurora and Barak, they give the following alternative definition of $BPP$:

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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 $Pr$?

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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.

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