Probability that one random number is larger than other random numbers
Suppose we have several urns, each containing some unknown number of balls, each of which is either (1) black with no number written on it or (2) white with some number written on it. Within each urn, the numbers on the white balls have a normal distribution with known mean and standard deviation (different for each urn). We will randomly select one ball from each urn and call an urn the winner if and only if the ball drawn from that urn has a higher number than the ball drawn from any other urn. (If all the balls drawn are black, there is no winner.) What is the probability that a given urn will be the winner?
I forgot to add: within each urn, we know the ratio of white to black balls.