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It is a bridgehead system with h = 4 fixed numbers and p – h = 2 variable numbers. The last 2 places of the lines were filled by unfolding all the 2-size combinations of the numbers from 5 to 49. This system contains C6−4 = C2 = 990 simple lines. Any equivalent 49−4 45 system obtained through a cyclic permutation of the 49 numbers will have the same parameters and number of simple lines. Let us observe that as the number of the variable numbers increases, the number of constituent simple lines increases much faster. For example, still for the 6/49 matrix, a bridgehead system with 3 variable numbers will have C6−3 = C3 =15180 simple lines, 49−3 46 while for 4 variable numbers, the system will have C6−2 = C4 =178365 simple lines, which, practically speaking, 49−2 47 makes it inapplicable. The bridgehead systems have the advantage of quickly checking the winning numbers and establishing the winning categories after the drawing, as well as the possibility of achieving multiple winnings, as we shall see further. Winning probabilities In this section, we refer to the cumulated winning probabilities for the bridgehead systems. Denoting by Bk the event “we will have minimum k winning numbers in at least one simple line of the system,” we aim to evaluate P(Bk ) . We have two cases: .............. missing part ................... Below, we present tables with numerical values of the probabilities returned by the previous general formula, for a large range of classical lottery matrices (in which p = n). We built a table for each value of n. For a given lottery matrix, the tables display the cumulated winning probabilities for any type of played bridgehead system, as a function of the number h of its fixed numbers. In a table, at the intersection of the row corresponding to a value of m with the 50PDF Image | THE MATHEMATICS OF LOTTERY Odds, Combinations, Systems
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