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In this paper we present a simple derivation of the probability of attaining a run of r heads when tossing a coin n times. We use this result to establish other properties of coin tossing. Using the same techniques we find the probability of runs of either heads or tails. We give a complete analysis of both recurrences. In addition we construct the generating functions for the recurrences.doi:10.4310/cis.2016.v16.n1.a3 fatcat:65r7tisslze2hahonjxewekhaq