Summary: A simple parallel randomized algorithm to find a maximal independent set in a graph \(G = (V, E)\) on \(n\) vertices is presented. Its expected running time on a concurrent-read concurrent-write PRAM with \(O(|E|d_{\max})\) processors is \(O(\log n)\), where \(d_{\max}\) denotes the maximum degree. On an exclusive-read exclusive-write PRAM with \(O(|E|)\) processors the algorithm runs in \(O(\log^2n)\). Previously, an \(O(\log^4n)\) deterministic algorithm was given by \textit{R. M. Karp} and \textit{A. Wigderson} [Proc. 16th Ann. ACM Symp. Theory of Computing, STOC 1984, 266--272 (1984); see also J. Assoc. Comput. Mach. 32, 762--773 (1985; Zbl 0633.68026)] for the EREW-PRAM model. This was recently (independently of our work) improved to \(O(\log^2n)\) by \textit{M. Luby} [Proc. 17th ACM Symp. Theory of Computing, STOC 1985, 1--10 (1985); see also SIAM J. Comput. 15, 1036--1053 (1986; Zbl 0619.68058)]. In both cases randomized algorithms depending on pairwise independent choices were turned into deterministic algorithms. We comment on how randomized combinatorial algorithms whose analysis only depends on \(d\)-wise rather than fully independent random choices (for some constant \(d)\) can be converted into deterministic algorithms. We apply a technique due to \textit{A. Joffe} [Ann. Probab. 2, 161--162 (1974; Zbl 0276.60005)] and obtain deterministic construction in fast parallel time of various combinatorial objects whose existence follows from probabilistic arguments.