(;) Semicolon Math: p-adic valuation

For all primes \(p\) and positive integers \(n\), denote \(v_p(n)\) by the greatest integer \(k\) such that \(p^k|n\). A nonnegative integer \(n\) is chosen at random from \(S=\{0, 1, \ldots, 2^{2021}-1\}\). The expected value of \(v_2\left(\binom{2n}{n}\right)\) can be expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\).