Background: Shannon's channel coding theorem says reliable communication is possible precisely when the channel's capacity exceeds the source's information rate. For a Binary Symmetric Channel (BSC, a channel that independently flips each transmitted bit with a fixed crossover probability ε), the capacity is C = 1 − Hb(ε) bits/use, where Hb(ε) = −εlog₂ε − (1−ε)log₂(1−ε) is the binary entropy function. For a binary source with entropy H(U) (its average information content, in bits/symbol), reliable transmission requires C > H(U). Because Hb(ε) is symmetric about ε = 0.5 and equals zero at both ε = 0 and ε = 1, this reliability condition is satisfied both for very clean channels and for channels that flip almost every bit — a channel that is "wrong" nearly all the time is just as informative as one that is right nearly all the time.
Description of This Web Application: Adjust the source probability P(X=0) and watch its entropy H(U) update live. The app numerically solves for exactly which ranges of the crossover probability ε make the BSC capacity exceed H(U), and shades those reliable regions directly on the Hb(ε) curve. By experimenting with different source distributions, you will see how source entropy and channel capacity interact, why the reliable region always appears in two symmetric bands near ε = 0 and ε = 1, and how a "noisier" source demands a cleaner channel to communicate reliably.