maths question help please

hi, please could i have help on the last part? i worked out the probability s=n is 2/3 because i thought the p(s)+p(x=1 or 2)=1 but not sure how that helps or if its correct?
the answer for n is 15, thanks!

Working out the probability is overthinking it a little, look at the ranges of the variables you've got in the question.
You're directly given that X, Y = 1, 2, 3, 4, 5, 6 (the values of a die).
Use the definition of S to consider which values of S are possible or not possible when X = 1 or 2 (and hence when the conditions are mutually exclusive).

sorry, im still a bit confused. When they are mutually exclusive, the two events wont overlap so does that mean we only consider x>3 s=x+y? I'm not sure what to do after that? thanks!

Your definition of mutual exclusivity is correct.
Having 'no overlap' means you want to find an n so that there are no cases (i.e., no combinations of X and Y) where {X = 1 or 2} and {S=n} are both true.
If you know that this only holds if X = 1 or 2, which part of the definition of S is relevant?
Then if X = 1 or 2, what values could S take?
And once you know the values which S could take, can you find an n which is not one of those values?