Functions arising by coin flipping

We repeatedly toss a biased coin where the probability is p of observing “heads”. The probability of a certain event can be regarded as a function of p. L. Lovász asked for a characterization of the functions occurring as the probability of some “reasonable” event. We state this problem in precise terms by introducing the concept of an “observable” event. Our main theorem states that a function (0, 1) → (0, 1) represents the probability of an observable event if and only if it is continuous, and does not tend to 0 or 1 faster than polynomially, as p tends to 0 or 1. We also give several simple examples of events having probabilities given by elementary functions.