Binary Symmetric Channel and the Repetition Code

What you are seeing: the Binary Symmetric Channel (BSC) is the simplest noisy channel: a transmitted bit flips with probability $p$ , stays with probability $1 - p$ . Shannon's noisy-channel theorem gives capacity $C(p) = 1 - H(p)$ where $H(p) = -p \log_2 p - (1-p) \log_2 (1-p)$ is the binary entropy. $C = 0$ at $p = 0.5$ and $C = 1$ at $p = 0$ or $1$ .

The top panel plots $C(p)$ and $H(p)$ . The bottom panel plots the bit-error-rate of an $n$ -repetition code (transmit each bit $n$ times, decode by majority vote) for $n = 1, 3, 5, 7, 11$ . Repetition trades rate for reliability: the error drops fast with $n$ at fixed $p \lt 0.5$ , at the cost of $n$ channel uses per source bit.

Figure 1. BSC capacity and repetition reliability. Method: closed-form entropy and binomial-sum formulas.

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WHAT TO TRY

Slide the flip probability p toward 0.5: the capacity C = 1 - H(p) collapses to zero. At p = 0.5 the output is independent of the input and the channel carries nothing.
Push p to 0 or 1: capacity returns to 1 bit per use. A channel that always flips is as good as one that never does, since the flip is deterministic and invertible.
Watch bits stream through: the fraction that flip matches p, and the capacity readout is the maximum rate you could still decode reliably with coding.