p(x) in Gaussian mixture model I've recently learned about Gaussian mixture models (GMM) in school. The formula for a GMM goes something like this:
$$p(x) = \sum_{k=1}^{K} \pi_i\mathcal{N}(x|\mu_i,\,\sigma^{2}_i)$$
Now, I know from probability theory that given a certain sample $x$, the probability for it should sum to 1.
For example, if I roll a six-sided die, the probability that I either get or don't get the number $5$ is equal to 1.
So what exactly does $p(x)$ here represent? The probability that a certain $x$ falls in the model? I feel like I'm missing something fundamental here.

The formula says that the random variable $X$ in question has probability density function $p(x)$ given by the weighted sum of $K$ different Gaussian densities, with weights summing to one.
The rationale behind a GMM is that, somehow, we believe your observations come from $K$ different groups, with each group having its own distribution. Then you leave the liberty to each observation to belong to any of each group. Thus you can think of data generated by a mixture model in two steps:
(1) draw group $i$ with probability $\pi_i$,
(2) draw an observation from the distribution of the group in (1).
Novices typically confuse the distribution of the mixture model with that of the sum of random variables, say $X_1+X_2+\cdots+X_K$. But, for instance, a sum of $K$ Gaussian random variables, $X_1+\cdots+X_K$, is still a Gaussian, whereas a random variable whose density is the sum of Gaussian densities (as the case in question) is not Gaussian.
