Sandbox
Mike Pearson provided a number of animations for these two articles. For the lottery example you can watch the numbers as they are chosen and produces a histogram for the number of times each number was chosen. When it finishes it looks like the number 38 was chosen significantly more times than any other number. A second animations allows you to see gaps between each time a number is chosen. You will see that the longest gap observed is 72 for the number 17 which seems surprising. The probability that a particular number would occur 72 times in a row is estimated to be about 1 in 12,5000.
The article then goes on to discuss how finding these apparent surprises does not necessarily indicate that the numbers were non randomly chosen. For the number of times a number occurs the authors show that the distribution for the number of times a number should occur in n draws has a binomial distribution with mean
They use a similar analysis to show that a gap of 72 is not too surprising. However, here they change the question.They calculate that in 1240 draws, each having 6 number, there are 7440 gaps that aspear. Then they ask:"What is the chance that the longest of these 7440 gaps is at lest 72". Using this they find there is about a 50% chance that a gap of at least 72 would appear.
Finally they discuss the use of a Chi Squared statistic to see if the approximate distributions from the data are consistent with the theoretical distributions used.