In this paper, we use random-selection protocols in the full-information model to solve classical problems in distributed computing. Our main results are the following: • An O(log n)-round randomized Byzantine Agreement (BA) protocol in a synchronous full-information network tolerating t < n 3+ faulty players (for any constant > 0). As such, our protocol is asymptotically optimal in terms of fault-tolerance. • An O(1)-round randomized BA protocol in a synchronous full-information network tolerating t = O( n