We will explore the introductory techniques of (arguably) one of the most influential mathematical developments of the past century: a sophisticated framework for solving general mathematics problems using artificial applications of randomness (i.e., coin flipping). The focus of this class will be on problems in discrete mathematics, particularly in the construction of mathematical structures with unexpected (and sometimes paradoxical) properties for which no other methods of construction are known.  However the underlying techniques have found numerous applications in almost all areas of mathematics and computer science.