Examples of nonexceedance probabilities in random sampling. (top) There are three people in a room (n = 3) ranked in order from the shortest (m = 1) to the tallest (m = 3). The probability that a person randomly drawn from this sample is not taller than the second tallest (m = 2) of them is 2/3 = m/n. However, the probability that a person randomly drawn from the population of all persons is not taller than the second-tallest person in the room is 1/2 = m/(n + 1). This can be seen by imagining that (bottom) this randomly chosen person has entered the room. Upon reranking, this person may fall in any of the new ranks first, second, third, and fourth, in two of which (first and second) the person is not taller than the second-tallest person in the original sample.
Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2533.1