etermine the transition matrix of a channel obtained by cascading two identical binary symmetric channels, and show that the result is also a binary symmetric channel. 12.3 (B): A binary channel successfully transmits bit 0 with a probability of 0.6 and bit 1 with a probability of 0.9. Assuming an input distribution P(X) = (q,1-q) with q = 0.2, determine the output source entropy H(Y). 12.4 (M): A binary symmetric channel with noise parameter E is used to transmit code- words of five-bit length. Determine the probabilities of transmitting codewords with: (a) One error; (b) Two errors; (c) At least two errors; (d) One burst of two successive errors. Application: provide in each case the numerical result, assuming that E = 0.1. Channel capacity and coding theorem This relatively short but mathematically intense chapter brings us to the core of Shannon's information theory, with the definition of channel capacity and the subse- quent, most famous channel coding theorem (CCT), the second most important theorem from Shannon (next to the source coding theorem, described in Chapter 8). The formal proof of the channel coding theorem is a bit tedious, and, therefore, does not lend itself to much oversimplification. I have sought, however, to guide the reader in as many steps as is necessary to reach the proof without hurdles. After defining channel capacity, we will consider the notion of typical sequences and typical sets (of such sequences) in codebooks, which will make it possible to tackle the said CCT. We will first proceed through a formal proof, as inspired from the original Shannon paper (but consistently with our notation, and with more explanation, where warranted); then with different, more intuitive or less formal approaches. Channel capacity In Chapter 12, I have shown that in a noisy channel, the mutual information, H(X;Y) = H(Y) - H(Y|X), represents the measure of the true information contents in the output or recipient source Y, given the equivocation H(Y|X), which measures the informationless channel nois