Law of Total Probability

Given event $A$, we have some event $A^c$, the complement of it. In particular,

$$\begin{array}{rl}{P}_X(x_i)& =P(X=x_i)\\ & =P(X=x_i,A)+P(X_i,A^c)\\ & =P_X(x_i\mid A)P(A)+P_X(x_i\mid A^c)P(A^c)\end{array}$$

If we partition $\Omega$ into $\{A_i{\}}_{i\in I}$, then $\Omega {\bigsqcup }_{i\in I}{A}_i$.

Continuous Variables

Given event $A$ and a random variable $X$ in which $X$ partitions $A$, then

$$\mathbb{P}(A)={\int }_{-\infty }^{\infty }\mathbb{P}(A\mid X=x)f_X(x)dx$$

Intuitively, since $X$ partitions $A$ (where $A$ depends on the value of $X$), it weights each of its own probabilities to $X$.

Joint Densities

Let $X,Y$ be two continuous random variables. Their joint density function is:

$$f_Y(y)={\int }_{-\infty }^{\infty }{f}_{X,Y}(x,y)dx$$

where we sum up all possible values of $X$ to find $Y$.

Expected Values

Read more here: Conditional Expectation.