The reasoning is correct, and fully mathematical. Probability is needed only for questions like "if I pick $32$ candies at random, what is the probability that I will have at least $15$ of the same colour?" (This is a lot harder.)
In explaining your answer, it may be useful to imagine, after you pick a bunch of candies, that you unwrap them and put them in boxes labelled green, red, and so on. Then we can use the sum of the number of candies in the various boxes to reason our way to a proof.
Formally, the proof consists of two parts: a) $50$ candies are not enough to ensure that we have $15$ of the same colour and b) $51$ candies are enough.
To prove a), we need only observe that the $50$ candies could consist of $14$ green, $14$ red, $12$ yellow, and $10$ blue. That was not explicitly stated in your argument, but it was implicitly there.
To prove b), observe that if we had no more than $14$ of any colour, then we could have at most $14+14+12+10=50$ candies. It follows that if the number of candies is $>50$, there will be at least $15$ of the same colour.
The two arguments a) and b) are logically quite distinct. However, they are so closely related that it is tempting (and not unreasonable) to prove a) and b) by a single piece of reasoning. In more complex situations, the arguments for "$A$ implies $B$" and "$B$ implies $A$" can be very different from each other. Indeed, the implication in one direction may be true, while the other implication is false. It is sometimes difficult to train students to distinguish between "$A$ implies $B$" and "$B$ implies $A$," since in most of the results they have seen in school, the implication runs in both directions.