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We study two polynomial counting questions in arithmetic statistics via a combination of Fourier analytic and arithmetic methods. First, we obtain new quantitative forms of Hilbert's Irreducibility Theorem for degree |$n$| polynomials |$f$| with |$\textrm {Gal}(f) \subseteq A_n$|⁠. We study this both for monic polynomials and non-monic polynomials. Second, we study lower bounds on the number of degree |$n$| monic polynomials with almost prime discriminants, as well as the closely related problem of lower bounds on the number of degree |$n$| number fields with almost prime discriminants. [ABSTRACT FROM AUTHOR]