MAT Exam 2021 Paper

Concept:Use the law of total probability to combine the two cases.Explanation:Probability of choosing the first bag when dice shows 1 or 3 is $P(B_1)=\frac{2}{6}=\frac{1}{3}$.Probability of choosing the second bag when any other face appears is $P(B_2)=\frac{4}{6}=\frac{2}{3}$.Probability of a black ball from the first bag is $P(K|B_1)=\frac{3}{7}$.Probability of a black ball from the second bag is $P(K|B_2)=\frac{4}{7}$.Total probability of choosing a black ball is:$$P(K)=P(B_1)P(K|B_1)+P(B_2)P(K|B_2)$$$$=\frac{1}{3}\times\frac{3}{7}+\frac{2}{3}\times\frac{4}{7}=\frac{3}{21}+\frac{8}{21}=\frac{11}{21}$$Answer:$\frac{11}{21}$, which is Option A.