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winning probability would become higher than 1. The error stands in the fact that the system with over 1033 lines will no longer obey the exclusiveness condition, through the increase of their common numbers, and the winning probability is given by formula (*). Depending on parameters a and c, players may establish their own probability thresholds for the played systems. The previous parameters, along with the chosen probability threshold, will provide through a simple calculation the minimal number of simple lines required for the system to offer the minimal probability chosen as threshold. But this number must be also limited by ensuring the profitability of the game (the eventual prize to be higher than the investment in the playing tickets). Denoting by pb the winning probability of a simple line in the category to which the prize of amount b is allocated, and by p* the chosen winning probability threshold of the system at the respective p* category, the above condition writes: p ⋅ c < b (**) b For a given lottery matrix and location, in relation (**) parameters b, pb and c can be considered constant, with an approximation for b as an average. (The prize fund and implicitly the amount of the prize allocated to a winning category changes from one game to another, depending on the ticket sales; the value of b can be calculated as the arithmetic mean of the amounts of the prizes in the respective category, as result of consulting the statistical history of the games on a determined previous period). p* The ratio p represents in fact the growing coefficient of the b initial winning probability pb (for one line), and its integer, approximated by addition, represents the minimum number of lines the system must contain for ultimately obtaining the winning probability p* . All the conditions mathematically expressed previously also stand for the events of cumulated winning in several categories (events of type “minimum k winning numbers”). In this case, pb will represent the cumulated winning probability for all considered categories. 37PDF Image | THE MATHEMATICS OF LOTTERY Odds, Combinations, Systems
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