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Assume that one aims to simulate an event of unknown probability $s\in (0,1)$which is uniquely determined, however only its approximations can be obtainedusing a finite computational effort. Such settings are often encountered instatistical simulations. We consider two specific examples. First, the exactsimulation of non-linear diffusions, second, the celebrated Bernoulli factoryproblem of generating an $f(p)-$coin given a sequence $X_1,X_2,...$ ofindependent tosses of a $p-$coin (with known $f$ and unknown $p$). We describea general framework and provide algorithms where this kind of problems can befitted and solved. The algorithms are straightforward to implement and thusallow for effective simulation of desired events of probability $s.$ In thecase of diffusions, we obtain the algorithm of \cite{BeskosRobertsEA1} as aspecific instance of the generic framework developed here. In the case of theBernoulli factory, our work offers a statistical understanding of theNacu-Peres algorithm for $f(p) = \min\{2p, 1-2\varepsilon\}$ (which is centralto the general question) and allows for its immediate implementation thatavoids algorithmic difficulties of the original version.

Simulating Events of Unknown Probabilities via Reverse Time Martingales