Eight data bits plus one even parity bit, 9 bits in total, are sent. If each bit is in error independently with a probability of 1%, which of the following is closest to the probability of an error that the parity check cannot detect?

Answer: B. About 0.34%

A parity check detects an odd number of bit errors, but an even number of bit errors leaves the number of 1s even or odd as before, so it cannot be detected. Almost all undetectable errors are 2-bit errors, whose probability is 9C2 × 0.01² × 0.99⁷ = 36 × 0.0001 × 0.932 ≈ 0.0034. The probability of errors in 4 or more bits is far smaller, so the total is still about 0.34%.

• A. 0.01² = 0.0001 is the probability that two particular bits are in error; it is not multiplied by the 36 ways of choosing 2 of the 9 bits.
• B. Correct. 9C2 × 0.01² × 0.99⁷ ≈ 0.34%.
• C. 1 − 0.99⁹ ≈ 8.6% is the probability of an error in at least one bit, which includes errors that parity can detect.