Team Guessing with Lacunary Information

Let E be an event of probability q and let U1,..., Un be independent random variables. There are n observers, the ith observing the n-1 random variables other than Ui. Each observer must guess whether E occurred. Then if pi is the probability of error of observer i, one has $$\prod_{i-1}^n p_i + p \geq P^{n-1}$$ where p = maxq, 1-q, and this bound is sharp. One application is related to a coding theorem. The coding is used to reduce degradation when data transmission is split over n parallel channels, any one of which might have broken down. The result also has a more symmetric corollary which can be stated as a probabilistic inequality of an unusual type.