law of total probability

Law of total probability

Given a partition of the sample space $\Omega$ into $n$ disjoint events ($\Omega ={\bigcup }_{i=1}^n{B}_i,B_i\cap B_j=\varnothing$), the probability of an event $A$ is the sum of the probabilities of $A$ given each of the $B_i$, weighted by the probability of each $B_i$:

$$\begin{array}{r}P(A)=\sum _{i=1}^{n}P(A\mid B_i)\cdot P(B_i)\end{array}$$

$P(A)=P(A\mid B)P(B)+P(A\mid B^c)P(B^c)$