Probability when selecting from 2 populations with different medians Given differing median values for two populations, what is the probability that a randomly selected person from the higher-median population will have a higher value than someone from the lower-median population? (I think I need to assume the populations are of similar size)

The median is not enough information. It could even be more likely for the woman's income to be higher.
Consider the following example.

This figure is an example histogram for the income of the two populations "men" (blue) and "women" (red). Clearly, the median of men is higher than the one of women, yet if you randomly choose a (man, woman) pair, it is more likely that the woman is having a higher income.
