Consider a communication channel that uses a protocol defined by the Error Detection Uncertainty Theorem. Let $M$ be the set of all messages over the channel, and VsubM the set of valid messages as defined by the protocol. The error detection method maps $V$ to $C,$ where CsubM represents codewords.

If the total number of valid messages $(|P|)$ is $10,000$ and the total number of codewords $(|C|)$ is $12,000,$ and assuming errors are uniformly distributed over the message space, calculate the probability $P($undetected$)$ that a randomly selected valid message is erroneously transformed into another codeword, remaining undetected by the protocol.