Maximum Entropy and the Lottery

The distribution on m-tuples of the first M integers is estimated from marginals of the distribution. This problem is of interest in determining unpopular numbers in lotto games. In Canada's Lotto 6/49 the proportion of tickets purchased in previous games containing each number is available. Under certain conditions the limiting distribution subject to the observed marginals is the constrained maximum entropy distribution. This distribution is estimated, and Monte Carlo methods are used to estimate the expected return of various lottery strategies. Tickets consisting of unpopular numbers may have expected return greater than their cost when the weekly sales are large or there are large carryover prizes (prizes not won in earlier games). The maximum entropy distribution is a rough approximation of the true distribution of tickets purchased. Certain aspects of the empirical distribution are not consistent with the maximum entropy distribution. Alternative methods, which attempt to model the behavior of ticket buyers, are considered.