χ2 and the lottery

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 χ 2 -distribution, even for large samples of draws. In fact, it can be shown that its asymptotic distribution is that of a weighted sum of χ 2 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 20 years.