Law of Total Probability

probability-theory

Definition

Law of Total Probability

If only the conditional probabilities and the probabilities of the conditional event are known, then the total probability of $A$ results from:

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

For event $A$ to occur, it must happen either with event $B$ or without event $B$. There are no other possibilities.

Partition of the sample space (general case): Given a set of events $B_1,B_2,B_3,\dots$ that form a partition of the sample space $\Omega$. A partition must satisfy two conditions:

1. Pairwise disjoint: The events do not overlap, i.e.: $\forall i\ne j:B_i\cap B_j=\varnothing$.
2. Collectively Exhaustive: The events cover the entire sample space, i.e. ${\bigcup }_{i=1}^{\infty }{B}_i=\Omega$.

If these conditions are met, the probability of any event $A$ is the sum of its conditional probabilities across all the partitions, weighted by the probability of each partition:

$$P(A)=\sum _{i=1}^{\infty }P(A\cap B_i)=\sum _{i=1}^{\infty }P(A\mid B_i)\cdot P(B_i)$$