\(\textbf{BPP}\) can now be defined to include the problems \(X\) such that there exists a probabilistic Turing machine \(C \in \mathfrak{C}\) and a constant \(\frac{1}{2} \lt p \leq 1\) with the following properties: \(C\) runs in polynomial time for all inputs; for all inputs \(x \in X\), at least fraction \(p\) of the possible computations of \(C\) on \(x\) accept; for all inputs \(x \not\in X\), at least fraction \(p\) of the possible computations of \(C\) on \(x\) reject. e. with probab