The Law of Total Probability

Study guide: Exam P-1

The Law of Total Probability states that if $\{A_1, \ldots, A_n\}$ is a partition of the sample space $S$ , then the probability of any event $B$ can be expressed as a weighted average of its conditional probabilities given each $A_i$ .

$\{A_1, A_2, \ldots, A_n\}$ must be mutually exclusive and exhaustive events to partition $S$
Used for computing the denominator $P(E)$ in Bayes Theorem and finding marginal probabilities when a problem is structured by distinct scenarios or risk classes
Every term equals $P(B \mid A_i)\,P(A_i) = P(B \cap A_i)$

$P(B) = \sum_{i=1}^{n} P(B \mid A_i)\,P(A_i)$

Overall Claim Probability Across Risk Classes A portfolio is 30% young drivers ( $A_1$ ), 50% middle-aged ( $A_2$ ), and 20% senior ( $A_3$ ). Claim probabilities are $P(C \mid A_1)=0.20$ , $P(C \mid A_2)=0.10$ , $P(C \mid A_3)=0.15$ . Find the overall probability of a claim.

Answer Since $\{A_1, A_2, A_3\}$ partitions the portfolio: $P(C) = (0.20)(0.30) + (0.10)(0.50) + (0.15)(0.20)$ $= 0.060 + 0.050 + 0.030 = 0.140$ The overall claim probability is 14%.