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Programs with randomization constructs is an active research topic, especially after the recent introduction of martingale-based analysis methods for their termination and runtimes. Unlike most of the existing works that focus on proving almost-sure termination or estimating the expected runtime, in this work we study the tail probabilities of runtimes-such as "the execution takes more than 100 steps with probability at most 1%." To this goal, we devise a theory of supermartingales thatarXiv:1811.06779v2 fatcat:lgchqbavufc6flxo6nn5tlp37q