Question

A (7,4) Hamming code is used over a BSC with flip probability $\epsilon = 0.1$. A transmitted codeword is decoded correctly if at most one bit error occurs. Find the probability of correct decoding (rounded to two decimals).

Hint:

For Hamming codes, decoding succeeds if the channel introduces 0 or 1 bit errors—use binomial probabilities.

Answer 1

Correct Answer: 0.84

A (7,4) Hamming code corrects up to one error. Let $\epsilon = 0.1$, so $(1-\epsilon) = 0.9$. Correct decoding occurs if:
- 0 errors, or
- 1 error.
Probability of 0 errors: \[ P_0 = (0.9)^7. \] Probability of 1 error: \[ P_1 = \binom{7}{1}(0.1)(0.9)^6 = 7(0.1)(0.9)^6. \] Compute numerically: \[ (0.9)^7 = 0.4783, \] \[ (0.9)^6 = 0.5314, \] \[ P_1 = 7(0.1)(0.5314) = 0.3720. \] Thus total correct-decoding probability: \[ P = P_0 + P_1 = 0.4783 + 0.3720 = 0.8503. \] Rounded to two decimals: \[ \boxed{0.85} \]