A persuasive coin is a sufficiently unbiased source of randomness visible to sufficiently many processors in a distributed system. An algorithm is described for achieving a persuasive coin in the presence of an extremely powerful adversary where the number of rounds of message exchange among the processors is constant, independent of the number n of processors in the system as well as the number of faults, provided the total number of faulty processors does not exceed a certain constant multiple of $n/\log n$. As a corollary an $\Omega (n/\log n)$-resilient probabilistic protocol for Byzantine agreement running in constant expected time is obtained. Combining this with a generalization of a technique of Bracha, a probabilistic Byzantine agreement protocol tolerant of almost ${n / 4}$ failures with $O(\log \log n)$ expected running time is obtained.