law of total probability

Let ${A_{1}, A_{2}, \dots}$ be a set of events that are mutually exclusive and collectively exhaustive.
For any event $E$ ,

\begin{aligned} P (E) & = \sum_{i} P (E \cap A_{i}) \\ = \sum_{i} P (E ∣ A_{i}) \cdot P (A_{i}) \end{aligned}

As a special case, consider the conditioning event $F$ and its complement $F^{∁}$ .
For any event $E$ ,

\begin{aligned} P (E) & = P (E \cap F) + P (E \cap F^{∁}) \\ = P (E ∣ F) \cdot P (F) + P (E ∣ F^{∁}) \cdot P (F^{∁}) \end{aligned}

Let ${A_{1}, A_{2}, \dots}$ be a set of events that are mutually exclusive and collectively exhaustive.

Since events $A_{i}$ are collectively exhaustive, for any event $E$ we have

E = E \cap Ω = E \cap (⋃_{i} A_{i})

E = ⋃_{i} (E \cap A_{i})

which is a union of disjoint sets since events $A_{i}$ are mutually exclusive.

P (E) = \sum_{i} P (E \cap A_{i})

and by the definition of conditional probability,

P (E) = \sum_{i} P (E ∣ A_{i}) \cdot P (A_{i})

QED.