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and of 5 numbers has 0.00287%. By contrast, the occurrence of 2 numbers has the probability 5.63%, the occurrence of one single number has 33.10%, while the occurrence of not having any winning number increases to over 60%! This phenomenon can be observed in every lotto matrix. Following any row of any table, we see the probabilities showing a sudden decrease in the last columns. In the 6/49 matrix, this jump occurs from w = 3 up, the probability decreasing about 18 times from w = 3 to w = 4, 53 times from w = 4 to w = 5 and 257 times from w = 5 to w = 6. For all matrices, the zone of the winning categories (with respect to w) correlates closely with the zone of these probability jumps. Let us also observe that probability does not decrease with w (by fixing the other parameters), so the maximal probabilities are not always obtained from the minimal values of w. For example, in a Keno game with p = 19, the maximal probability is obtained when w = 5. In a 7/47 game, the maximal probability is obtained when w = 1, while in the 6/49, maximal probability is obtained when w = 0. Cumulated winning probabilities In the previous section, we presented the probabilities of occurrence of a fixed number of numbers from the played line, in the draw (events denoted by Aw ). We saw that these probabilities are very low for superior values of w. Within the same lottery matrix, a player will be also interested in the cumulated winning probabilities corresponding to the occurrence of a variable number of played numbers in the draw. This relates to the events of type “at least w numbers.” Although the mathematical information offered by the probabilities of these compound events is included in the results within the previous section, the new cumulated results provide us with an evaluation and decision criterion that is more practical. Staying with the same denotations for the parameters, we want to evaluate the probability of occurrence of at least w numbers from the p of the played line, in the draw: 1, 2, ..., p – 1 (the case “at least 27PDF Image | THE MATHEMATICS OF LOTTERY Odds, Combinations, Systems
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