The Equivalency of Two Definitions of Completeness & Soundness in Interactive Proof Systems
The prover is "all-powerful and possesses unlimited computational resources" so it has no need of random bits. Thus the only randomness is the randomness of the verifier.

If the prover uses random bits, it should replace them with the random bit string that is most likely to make the verifier accept (this is true for both the honest and any dishonest prover). Furthermore, the prover can determine this optimal bit string because the prover is "all-powerful".