In combinatorics on words, a word w of length n over an alphabet of size q is said to be privileged if n ≤ 1, or if n ≥ 2 and w has a privileged border that occurs exactly twice in w. Forsyth, Jayakumar and Shallit proved that there exist at least 2^n-5/n² privileged binary words of length n. Using the work of Guibas and Odlyzko, we prove that there are constants c and n₀ such that for n ≥ n₀, there are at least cqⁿ/(n(log_q n)²) privileged words of length n over an alphabet of size q. Thus, for n sufficiently large, we improve the earlier bound determined by Forsyth, Jayakumar and Shallit, and generalize it for all q.