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The number of prizes The fact that the simple lines corresponding to a compound line are generated by combining a limited number of given numbers implies an accumulation of common numbers in these lines, which we generally do not see in random systems, and even less frequently in systems having the exclusiveness property. Due to this combinatorial accumulation, a compound line may ensure – in case of winning in an arbitrary category – the simultaneous existence of several winnings in lower categories. Next, we deduce a general formula that returns the number of possible winnings of a compound line, for every lottery matrix, as a function of the size of the compound line and its number of winning numbers. .............. missing part ................... Example of calculation: In a 6/49 matrix, from a 12-number compound line, 5 numbers are winning. How many simple lines containing exactly 5 and exactly 4 winning numbers are in that system? We have p = 6, r = 12, k = 5. We apply the above formula for i=5andi=4.Weobtain N5 (5)=C5C6−5+5 =C6 =7 lines 6,12 512−5 7 containing the 5 winning numbers and N5 (4) = C4C6−5+4 = C5 = 21 lines containing exactly 4 from the 5 6,12 512−5 7 winning numbers. This means that we will have simultaneously 7 prizes in the second category and 21 prizes in the third category. Next we present tables of values of number N k (i) , returned by p,r the general formula, for the most popular lottery matrices (each matrix being represented through a value of p) and for a large enough range of values for the size of the compound line. In these tables, in the first row are noted the values of the number of winning numbers (k), in the second row are values of the number of winning numbers that are less than or equal to k (namely k – i), and in the first column are the values of the size of the compound line (r). 45PDF Image | THE MATHEMATICS OF LOTTERY Odds, Combinations, Systems
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