Question

The average bit error rate at the input of a \((7,4,1)\) Hamming decoder is \(0.10\).
The probability that the decoder will fail to decode a received word correctly is (rounded off to two decimal places).

Hint:

The decoder fails only when 2 or more of the 7 bits are wrong, so subtract the 0-error and 1-error binomial probabilities from 1.

Answer 1

Correct Answer: 0.15

Step 1: Frame success instead of failure.
$P(\text{success}) = P(0 \text{ errors}) + P(1 \text{ error})$, since the code fixes up to one flipped bit. Let $q=1-p=0.9$.

Step 2: Factor out the common power of q.
\[ P(\text{success}) = q^7 + 7pq^6 = q^6(q+7p) \]
This factoring saves work because we only need $q^6$ once instead of computing $q^6$ and $q^7$ separately.

Step 3: Simplify the bracket.
\[ q+7p = (1-p)+7p = 1+6p = 1+6(0.1)=1.6 \]

Step 4: Compute $q^6$.
\[ q^6=(0.9)^6=0.531441 \]

Step 5: Multiply.
\[ P(\text{success})=0.531441\times1.6=0.850306 \]

Step 6: Get the failure probability.
\[ P(\text{fail})=1-0.850306=0.149694 \]
Rounded to two places, this is $0.15$.
\[ \boxed{0.15} \]