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Chi-Square and the Lottery Essay

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Chi-square and the Lottery
Christian Genest1 , Richard A. Lockhart2 and Michael A. Stephens2
de math´matiques et de statistique, Universit´ Laval, e e Qu´bec (Qu´bec), Canada G1K 7P4 e e 2 Department of Statistics and Actuarial Science, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 Summary The winners of many lotteries are determined by selecting at random some numbered balls from an urn. This paper discusses the use of Pearson’s standard goodness-of-fit statistic to test for the equiprobability of occurrence of such lottery numbers, whether taken individually, in pairs or in larger subsets. Because the numbers are drawn without replacement, Pearson’s statistic does not follow a simple chi-square distribution, even for large samples of draws. In fact, it can be shown that its asymptotic distribution is that of a weighted sum of chi-square random variables. An explicit formula is given for the weights, and this result is used to check the uniformity of winning numbers in Canada’s Lotto 6/49 over a period of nearly twenty years. Key words: Chi-square distribution; goodness-of-fit; Johnson association scheme; lottery; Lotto 6/49; Pearson’s statistic. Submitted 23 October 2001 Revised 28 December 2001
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1 Introduction A large number of countries of the world run lotteries, and several sub-national political entities (states, provinces, territories, etc.) do as well. A common way in which many of these lotteries are run is as follows. The gambler picks k distinct numbers from the set {1, . . . , N }, and then pays a fixed amount to register the selection with an authorized agent. Alternatively, the gambler may prefer to purchase a “QuickPick” ticket whose random subset of k numbers is generated automatically using an algorithm devised by the Lottery Corporation. At the close of an alloted time period, the winning combination is determined by drawing at random k balls without replacement from an urn containing identical balls...

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