In a binary symmetric channel, 0s and 1s are sent and each bit flips independently with probability 0.1. With the binary entropy function H(p) = −p log₂ p − (1 − p) log₂(1 − p), H(0.1) ≈ 0.469. Which of the following is closest to the channel capacity of this channel per transmission?

Answer: B. About 0.53 bits

The channel capacity of a binary symmetric channel is C = 1 − H(p), the value of the mutual information when 0 and 1 are sent with equal probability. For p = 0.1, C ≈ 1 − 0.469 = 0.531, about 0.53 bits per transmission. With suitable error-correcting codes, data can be sent at up to this rate with an arbitrarily small error rate. At p = 0.5, the received value is unrelated to the value sent, and the capacity is 0.

• A. About 0.47 bits is H(0.1), the uncertainty caused by the flips. The capacity is 1 minus this value.
• B. Correct. 1 − H(0.1) ≈ 0.53 bits.
• C. 0.9 is the fraction of bits that arrive correctly, but the receiver does not know which bits flipped, so that information is lost and the capacity is smaller than 0.9.