I'm trying to find out whether my communication protocol should have redundant information padded, in order to help the receiver correct the error (error correction code, ECC) without needing a retransmission. In other words, I'm trying to find the point at which it's more interesting to have a small overhead for each message, rather than a whole message retransmission from time to time. If needed, I assume I can ask God to tell me if the packet is altered or not.
Let's take a ECC for which, when t extra bits added, t/2 corrections are possible. Each bit has an error rate of p, so the probability P that the message gets altered (assuming independent errors) is $P=1-(1-p)^n$, where n is the message length. The probability that less than t/2 error occurs is: $\displaystyle P(X\leq\frac t2)=\sum_{k=0}^{\lfloor\frac t2\rfloor}{n+t\choose k}p^k(1-p)^{n+t-k}$ I think I have done most of the work but I'm not sure how to go from there to solve my initial question. I think I will need to solve the inequation $P\leq P(x\leq\frac t2)$, but I'm not sure, and in any case I don't know how to solve it. If it turns out to be infeasible to solve, I can give numerical values to help with the resolution.