The image describes the Law of Total Probability. Given a sample space $S$ of an experiment $E$, and an event $A$ in $E$. Suppose $B_1, B_2, ..., B_n$ form a partition of $S$, with $P(B_i) > 0$ for $i=1, 2, ..., n$. The problem presents the formula for the probability of event $A$, $P(A)$.

1. Problem Description

The image describes the Law of Total Probability. Given a sample space $S$ of an experiment $E$, and an event $A$ in $E$. Suppose $B_1, B_2, ..., B_n$ form a partition of $S$, with $P(B_i) > 0$ for $i=1, 2, ..., n$. The problem presents the formula for the probability of event $A$, $P(A)$.

2. Solution Steps

The Law of Total Probability states that if events $B_1, B_2, ..., B_n$ form a partition of the sample space $S$, then the probability of any event $A$ can be expressed as the sum of the probabilities of $A$ occurring with each of the $B_i$ events. This can be represented as:

$P(A) = P(A \cap B_1) + P(A \cap B_2) + ... + P(A \cap B_n)$.

Since $P(A \cap B_i) = P(A|B_i)P(B_i)$, the Law of Total Probability can be written as:

$P(A) = P(A|B_1)P(B_1) + P(A|B_2)P(B_2) + ... + P(A|B_n)P(B_n)$.

3. Final Answer

$P(A) = P(A|B_1)P(B_1) + P(A|B_2)P(B_2) + ... + P(A|B_n)P(B_n)$