Abstract
Summary Suppose two friends are about to participate in a two-day contest in which they repeatedly attempt a task with a clear success/failure outcome (such as shooting free throws on a basketball court). We have no specific prior knowledge of the participants’ skills, how the change in day will impact their success, or how many attempts they will take each day, so we suppose that each participant’s success rate for day 1, success rate for day 2, and proportion of attempts that take place on day 1 are all chosen uniformly at random between 0 and 1. What is the probability that the same person has a higher success rate each of the two individual days, but the other person has a higher success rate for the two-day period? This can be thought of as a prior probability of a simple case of Simpson’s paradox, and we show that this probability is ( π 2 − 9 ) / 36 = .0241556778 … .
Published Version
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