Marginalisation (probability)

Suppose we have two random variables $X$ and $Y$ over domains $A$ and $B$ respectively. If we know their join distribution $\mathbb{P}[X, Y]$ then we can calculate either $X$ or $Y$’s (marginal) distribution, i.e.

$$\mathbb{P}[X = a] = \sum_{b \in B} \mathbb{P}[X=a, Y=b] = \sum_{b \in B} \mathbb{P}[X=a | Y=b] \mathbb{P}[Y=b].$$

The name comes from if you were to draw a table we would get both $\mathbb{P}[X]$ and $\mathbb{P}[Y]$ in the margins if we summed the rows and columns.