How often will this test fail?
Wolfram Language (Mathematica), 32 bytes
#3~b~#/b[#2#3,#]#2^#&amp;
b=Binomial

Try it online!
Input [p, n, a].
The formula can be rewritten \$\dfrac{a\choose p}{an\choose p}n^p\$.

Incidentally, this is fairly trivial using Mathematica's statistics built-ins:
Wolfram Language (Mathematica), 69 bytes
CDF[MultivariateHypergeometricDistribution[#~Max~1,a=Table@##2],1^a]&amp;

Try it online!
Unfortunately, 38-character name aside (the longest name in the default namespace), MultivariateHypergeometricDistribution doesn't handle a nonpositive prefix length, and overall this approach is quite slow for this particular problem.