2015 DHS P2 Q7 [Modified]

Data is transmitted in bytes, where each byte consists of \(8\) bits. The probability of a bit being corrupted during its transmission is \(0.03\). A byte is considered ‘corrupted’ if it contains at least \(2\) corrupted bits.

Assume that all bits are not corrupted prior to their transmission.

1. Find the probability that a randomly chosen byte is corrupted during its transmission.
2. Given that a randomly chosen byte is not corrupted during its transmission, find the probability that it contains no corrupted bits.
3. Find the probability that between \(5\) and \(10\) bytes are corrupted during the transmission of \(100\) bytes.

In another transmission system, the probability of having at least \(3\) corrupted bytes during the transmission of \(20\) bytes is \(0.85\). Find the probability that a bit is corrupted during its transmission.