Dice games, part 2

Calculus Level 5

Let \(E_{n,p}\) be the expected value of rolling an \(n\)-sided die \(p\) times and choosing the greatest number shown.

The limit \( \displaystyle\lim_{n \to \infty} \frac{E_{n,20}}{E_{n,30}} \) can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are relatively coprime positive integers. Find \(a+b\).

Related to a curiosity I discovered while writing Dice game.
Image Credit: Wikimedia Pentakisdodecahedron.

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