Randomizing Sequences of Finite Length

유한 순서열의 임의화

It is never an easy task to physically randomize the sequence of cards. For instance, US 1970 draft lottery resulted in a social turmoil since the outcome sequence of 366 birthday numbers showed a significant relationship with the input order (Wikipedia, "Draft Lottery 1969", Retrieved 2009/05/01). We are motivated by Laplace's 1825 book titled Philosophical Essay on Probabilities that says "Suppose that the numbers 1, 2, ..., 100 are placed, according to their natural ordering, in an urn, and suppose further that, after having shaken the urn, to shuffle the numbers, one draws one number. It is clear that if the shuffling has been properly done, each number will have the same chance of being drawn. But if we fear that there are small differences between them depending on the order in which the numbers were put into the urn, we can decrease these differences considerably by placing these numbers in a second urn in the order in which they are drawn from the first urn, and then shaking the second urn to shuffle the numbers. These differences, already imperceptible in the second urn, would be diminished more and more by using a third urn, a fourth urn, &c." (translated by Andrew 1. Dale, 1995, Springer. pp. 35-36). Laplace foresaw what would happen to us in 150 years later, and, even more, suggested the possible tool to handle the problem. But he did omit the detailed arguments for the solution. Thus we would like to write the supplement in modern terms for Laplace in this research note. We formulate the problem with a lottery box model, to which Markov chain theory can be applied. By applying Markov chains repeatedly, one expects the uniform distribution on k states as stationary distribution. Additionally, we show that the probability of even-number of successes in binomial distribution with trials and the success probability $\theta$ approaches to 0.5, as n increases to infinity. Our theory is illustrated to the cases of truncated geometric distribution and the US 1970 draft lottery.

미국의 1970년 징병추출(draft lottery)은 유한 순서열 (1, 2, ..., k)의 물리적 임의화를 쉬운 일로 생각하였다가 사회적 물의가 빚어진 대표적인 사례이다. 본 소고는 숫자 1, 숫자 2, ... 등의 순서로 쌓인 k장의 카드 뭉치를 물리적으로 임의화하는 데 있어 반복 시행(repeated trial)의 역할을 밝힌다. 부수적으로 독립시행 수 n, 성공의 확률이 $\theta$인 이항분포 B(n, $\theta$)에서 성공 수가 짝수일 확률은 n이 커짐에 따라 0.5에 수렴하게 됨을 보인다.