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As an example, for a 6/49 lottery, Ω has 13,983,816 elements. For 6/42 lottery, Ω has 5,245,786 elements. For 5/40 lottery, Ω has 658,008 elements. For 3/90 lottery, Ω has 117,480 elements. We consider the field of events as being the set of parts of the sample space, so this set is also finite. As a set of parts of a set, the field of events is a Boole algebra. Any event belonging to the field of events, no matter how complex, can be broken down into units of elementary events, by using the axioms of Boole algebra. Because the events are identified with sets of numbers and the axioms of a Boole algebra, the operations between events (union, intersection, complementary) revert to the operations between sets. Therefore, any counting of elementary events (for example, the elementary events comprising a compound event) reverts to counting number combinations. Thus, the combinatorial calculus becomes an essential tool for the probability calculus applied in lottery. Examples of events: In the 6/49 lottery, the event occurrence of a combination containing numbers 3, 5, 7, 11, 15 is the set of elementary events A = {(3, 5, 7, 11, 15, 1), (3, 5, 7, 11, 15, 2), (3, 5, 7, 11, 15, 4), ...}, or A= (3,5,7,11,15,x),x∈{1,2,,49}−{3,5,7,11,15} . {} This set has 49 – 5 = 44 elements (6-number combinations). The event occurrence of a combination containing only numbers between 21 and 29 is the set of elementary events B = {(21, 22, 23, 24, 25, 26), (21, 22, 23, 24, 25, 27), ...} or B= (x,y,z,t,u,v),x,y,z,t,u,v∈{21,22,,29} (withx,y,z,t, {} u, v mutually distinct). This set has C96 = 84 elements. .............. missing part ................... 17PDF Image | THE MATHEMATICS OF LOTTERY Odds, Combinations, Systems
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