When Buffon's needle is dropped at random on parallel lines spaced by the needle's length, it crosses a line with probability 2/pi. Add another set of parallel lines, perpendicular to the first, and with the same spacing. The probability that the needle crosses two lines, one from each set, is not equal to (2/pi)^2, as it would if the two types of crossing were independent, but 1/pi. So there is the curious fact that the probability of crossing a horizontal line and a vertical line is equal to the probability of crossing a horizontal line and not crossing a vertical line (since their sum must be 2/pi). Equivalently, the conditional probabilities, of crossing or not crossing a vertical line, given that a horizontal line is crossed, are equal. If you believe there are no accidents in mathematics, then surely there is some bijection that explains this fact ...