Calculation of failure probability in basic Ring-LWE-DH key agreement
One concrete solution is to explicitly compute the distribution of each coefficients. In this particular ring, and if the distribution of the coefficients of e and s are symmetric, this ca just be computed as an 2n-fold convolution of the distribution of products of coefficients. An example (with some complication due to rounding) is available here:

https://github.com/pq-crystals/security-estimates/blob/master/Kyber_failure.py

More mathematical approaches (which may give somewhat looser bound) involve studying moment-generating function. This can be nicely abstracted away using the theory of sub-gaussian random variables.