The Hardy-Weinberg Law is about allelic independence, and may be stated as follows: In a closed random-mating population without selection, a progeny's alleles are independently sampled from the respective parental gene pools. Population genetics textbook proofs of this fundamental result have hardly changed since its discovery in 1908. The standard argument, whcih applies Mendel's First Law on mating types, presents significant algebraic challenge for multiple alleles; even for the biallelic case, it takes quite a bit of concentration to follow. It is observed that the Law is a sort of counter-example to Simpson's paradox. This key observation inspires a new simple argument that retains its elegance for any number of alleles. Thus, it should be incorporated in standard textbooks. Furthermore, it is found that random mating together with certain fertility selection can result in allelic independence, which may have important implication on the interpretation of real data: allelic independence may not mean no selection. Also obtained is a mathematical characterisation of fertility selection coefficients that work a given population to yield allelic independence.