Let $X$ and $Y$ be independent random variables, each uniformly distributed on $\{1,2,3,\ldots,11\}.$ I want to find $\mathrm{P}(X+Y=16).$
Well, the joint probability mass function of $X$ and $Y$ is give by $ P(X=x, Y=y)=\frac{1}{11^2}.$ To compute the above probability, I find all combinations of $X$ and $Y$ whose sum is 16. Doing that I get 7 pairs, and hence $P(X+Y=16)=\frac{7}{11^2}.$
My question is this: Is there a better way of approaching these types of problems other than the procedure I've outlined above?