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: Understand what the code can and cannot fix.
A \((7,4,1)\) Hamming code sends a \(7\)-bit codeword built from \(4\) message bits, and the \(1\) means it can correct at most \(1\) bit error in each received word. If \(2\) or more bits in a \(7\)-bit word are flipped, the decoder either corrects to the wrong codeword or cannot fix the word at all. So the decoder fails exactly when \(2\) or more of the \(7\) bits are in error.

Step 2: Set up the bit error model.
Each of the \(7\) bits is wrong independently with probability \(p=0.10\) and correct with probability \(1-p=0.90\). The number of wrong bits in a \(7\)-bit word follows a binomial distribution with \(n=7\) trials.

Step 3: Write the failure probability.
\[ P(\text{fail})=1-P(0\text{ errors})-P(1\text{ error}) \]
because the decoder only succeeds when the word has \(0\) errors, or exactly \(1\) error that it can correct.

Step 4: Compute the probability of zero errors.
\[ P(0\text{ errors})=(1-p)^7=(0.9)^7 \]
\[ (0.9)^7=0.4783 \]

Step 5: Compute the probability of exactly one error.
There are \(7\) ways to choose which single bit is wrong, so
\[ P(1\text{ error})=\binom{7}{1}p(1-p)^6=7(0.1)(0.9)^6 \]
\[ (0.9)^6=0.5314 \]
\[ P(1\text{ error})=7(0.1)(0.5314)=0.3720 \]

Step 6: Combine and subtract from 1.
\[ P(0\text{ or }1\text{ error})=0.4783+0.3720=0.8503 \]
\[ P(\text{fail})=1-0.8503=0.1497 \]

Step 7: Round the answer.
Rounded to two decimal places, the probability of decoding failure is
\[ \boxed{0.15} \]