Mapping from a Single X to Typical Y Outputs
For any input block x ∈ Xn = {0,1}n, only ≈2(n·Hb(ε)) outputs Y are "typical" (high probability). The rest of Yn exists but is reached with vanishingly small total probability.
Background: Shannon's noisy-channel coding theorem relies on a joint version of the AEP (Asymptotic Equipartition Property): when a fixed input sequence xn is sent over n uses of a noisy channel (here, a Binary Symmetric Channel, or BSC, with crossover probability ε), the set of outputs it could plausibly produce concentrates into a "typical set" of size roughly 2nH(Y|X) (H(Y|X) = conditional entropy of the output given the input, in bits) — far smaller than the 2n total possible outputs. If codewords are chosen so their typical output sets barely overlap, the receiver can uniquely identify which codeword was sent, and counting how many such non-overlapping sets fit inside the output space shows that roughly 2nC codewords can be reliably distinguished, where C is the channel capacity.
Description of This Web Application: Adjust the block length n and crossover probability ε for a binary symmetric channel, then explore four linked views: how a single input sequence maps to its typical set of likely outputs, how many non-overlapping codewords can be packed into the output space, how the number of bit errors is distributed across n channel uses, and how channel capacity C = 1 − Hb(ε) (Hb = binary entropy function) varies with ε. By the end, you should have the geometric intuition behind why reliable communication is possible at any rate below capacity, and how increasing the block length shrinks the relative size of the typical output set.
For any input block x ∈ Xn = {0,1}n, only ≈2(n·Hb(ε)) outputs Y are "typical" (high probability). The rest of Yn exists but is reached with vanishingly small total probability.