10,000 bits of data are sent over a link with a bit error rate of 1 × 10⁻⁴. If the errors in each bit occur independently, which of the following is closest to the probability that at least one bit is in error?

Answer: B. About 63%

The probability that no bit is in error is (1 − 10⁻⁴)^10,000 ≈ e^(−1) ≈ 0.37, so the probability that at least one bit is in error is 1 − 0.37 = 0.63, or about 63%. When n × p is much smaller than 1, 1 − (1 − p)ⁿ can be approximated by n × p, but here n × p = 1, so this approximation cannot be used.

• A. 0.01% is the error rate per bit, not the probability that any of the 10,000 bits is in error.
• B. Correct. 1 − (1 − 10⁻⁴)^10,000 ≈ 1 − e^(−1) ≈ 0.63.
• C. This takes 10,000 × 10⁻⁴ = 1 as the probability, which counts the cases with two or more bit errors more than once. The n × p approximation can be used only when n × p is small enough.