Q.Out of the adult population in a village, 50% are farmers, 30% do business and 20% are service holders. It is known that 10% of farmers, 20% of businessmen and 50% of service holders are above poverty line. What is the probability that a villager chosen from the adult population of the village, selected at random, is above poverty line?

Theorem 12.10 (Total Probability of an Event). Suppose $A_1,A_2,\ldots,A_n$ are mutually exclusive AND exhaustive events (a partition of the sample space $S$), and $B$ is any event in $S$. Then

Why it works. Since $A_1,\ldots,A_n$ partition $S$, the event $B$ splits into the mutually exclusive pieces $(A_1\cap B),(A_2\cap B),\ldots,(A_n\cap B)$, whose union is $B$ itself. Adding their probabilities (axiom $[P_2]$) gives $P(B)=\sum_i P(A_i\cap B)$, and the Multiplication Theorem rewrites each term as $P(A_i)P(B/A_i)$. …