Why does a binary symmetric channel transmit no useful information when each bit has a 50 percent chance of being correct?
Consider a simple telephone conversation: A person (message source) speaks into a telephone receiver (encoder), which converts the sound of the spoken word into an electrical signal. This electrical signal is then transmitted over telephone lines (channel) subject to interference (noise). When the signal reaches the telephone receiver (decoder) at the other end of the line it is converted back into vocal sounds. Finally, the recipient (message receiver) hears the original message.
In a binary symmetric channel, the effectiveness of information transmission is determined by the channel's capacity to convey data after accounting for the bits required for error correction. This efficiency is calculated using the formula , where represents the average number of bits needed for error correction per transmitted bit.1
When the probability of a bit being received correctly is , the error correction requirement becomes 1. Substituting this into the efficiency formula results in an effectiveness of 0 percent. At this level of noise, the uncertainty is absolute; the error rate is so high that the received signal provides no reliable indication of the original input. Because there is no way to distinguish between a correct symbol and an error, the channel is effectively equivalent to flipping a coin at the receiving end, rendering it incapable of transmitting useful information.1