- A computer communication channel transmits words of $n$ bits using an error-correcting code which is
capable of correcting errors in up to $k$ bits. Here each bit is either a $0$ or a $1$. Assume each bit is
transmitted correctly with probability $p$ and incorrectly with probability $q$ independently of all other
bits.
(a) Find a formula for the probability that a word is correctly transmitted.
(b) Calculate the probability of correct transmission for $n = 8$, $k = 2$, and $p = 0:01$.
Best Answer
This is a typical Binomial Distribution calculation. The probability that $k$ or fewer bits are transmitted incorrectly is $$\sum_{i=0}^k \binom{n}{i}q^ip^{n-i}.$$
Remark: In the formula, I have used your assertion that the probability that a bit is transmitted incorrectly is called $q$. There may be a typo in the question, since in the concrete example you use $p=0.01$, meaning that the probability $q=1-p$ that a bit is transmitted incorrectly is $0.99$. If there is such a high probability of incorrect transmission of a bit, we would be much better off reversing the bit received!
If there really is a typo, and the probability of incorrect transmission is $p$, then in the formula above, just interchange $q$ and $p$.