ON THE RANDOM SAMPLING OF PAIRS, WITH PEDESTRIAN EXAMPLES

Abstract

Abstract. Suppose one desires to randomly sample a pair of ob-jects such as socks, hoping to get a matching pair. Even in the sim-plest situation for sampling, which is sampling with replacement, the innocent phrase “the distribution of the color of a matching pair ” is ambiguous. One interpretation is that we condition on the event of getting a match between two random socks; this corre-sponds to sampling two at a time, over and over without memory, until a matching pair is found. A second interpretation is to sam-ple sequentially, one at a time, with memory, until the same color has been seen twice. We study the difference between these two methods. The input is a discrete probability distribution on colors, describing what happens when one sock is sampled. There are two derived distri-butions — the pair-color distributions under the two methods of getting a match. The output, a number we call the discrepancy o