Balls are drawn one at a time from a bag containing 3 black balls, 4 red balls, and 5 white balls, and all 12 drawn balls are arranged in a horizontal row in the order they were drawn. Each ball in the bag is equally likely to be drawn.
[(1)] Find the probability $p$ that no two red balls are adjacent to each other.
[(2)] Given that no two red balls are adjacent to each other, find the conditional probability $q$ that no two black balls are adjacent to each other.
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Balls are drawn one at a time from a bag containing 3 black balls, 4 red balls, and 5 white balls, and all 12 drawn balls are arranged in a horizontal row in the order they were drawn. Each ball in the bag is equally likely to be drawn.
\begin{enumerate}
\item[(1)] Find the probability $p$ that no two red balls are adjacent to each other.
\item[(2)] Given that no two red balls are adjacent to each other, find the conditional probability $q$ that no two black balls are adjacent to each other.
\end{enumerate}
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