The marginal probability equation follows:
\begin{equation} \sum_{Y} P(X = x | Y = y)P(Y = y) \end{equation}
Practically, it seems that most computations actually utilize the sum of the conditional probabilities rather than the sum of joint probabilities,
\begin{equation} \sum_{Y} P(X = x | Y = y) \end{equation}
Given my lack of knowledge in the probability domain, is P(Y = y) always equal to 1, or are there any special cases where P(Y = y) would be less than 1? (In which case, computation would be explicitly performed using the sum of the joint probabilities).
For discrete random variables, $X,Y$, the Law of Total Probability states:
$$\begin{align}\mathsf P(X{=}x)~&=~\sum_y\mathsf P(X{=}x, Y{=}y)\\[2ex]&=~\sum_y\mathsf P(X{=}x\mid Y{=}y)~\mathsf P(Y{=}y)\end{align}$$
Note: the conditional probability $\mathsf P(X{=}x\mid Y{=}y)$ is not generally equal to the joint probability $\mathsf P(X{=}x, Y{=}y)$. Indeed, the definition for conditional probability is that for discrete random variables (where $\{Y{=}y\}$ is not a zero measure event) we have:$$\mathsf P(X{=}x\mid Y{=}y)~=~\dfrac{\mathsf P(X{=}x, Y{=}y)}{\mathsf P(Y{=}y)}$$