In Boolean synthesis, we are given an LTL specification, and the goal is to construct a transducer that realizes it against an adversarial environment. Often, a specification contains both Boolean requirements that should be satisfied against an adversarial environment, and multi-valued components that refer to the quality of the satisfaction and whose expected cost we would like to minimize with respect to a probabilistic environment.In this work we study, for the first time, mean-payoff games in which the system aims at minimizing the expected cost against a probabilistic environment, while surely satisfying an ω-regular condition against an adversarial environment. We consider the case the ω-regular condition is given as a parity objective or by an LTL formula. We show that in general, optimal strategies need not exist, and moreover, the limit value cannot be approximated by finite-memory strategies. We thus focus on computing the limit-value, and give tight complexity bounds for synthesizing ǫ-optimal strategies for both finite-memory and infinite-memory strategies. We show that our game naturally arises in various contexts of synthesis with Boolean and multi-valued objectives. Beyond direct applications, in synthesis with costs and rewards to certain behaviors, it allows us to compute the minimal sensing cost of ω-regular specifications -a measure of quality in which we look for a transducer that minimizes the expected number of signals that are read from the input.C is a GEC Pr(f reaches and stays in C) • v(C). Consider the strategy f as a strategy for M ′ . Then, cost M ′ (f ) = C is a GEC Pr(f reaches and stays in C)• v(C), and we conclude that cost sure (M) ≥ cost (M ′ ).For the other direction, we show that cost sure (M) ≤ cost (M ′ ). Since M ′ is an MDP, then there exists an optimal memoryless strategy f ′ such that cost M ′ (f ′ ) = cost (M ′ ). We show that for every ǫ > 0, there exists a winning strategy f for M such that cost M (f ) ≤ cost M ′ (f ′ ) + ǫ.