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Winning probabilities For evaluating the probability of occurrence in the draw of a certain number of winning numbers in the case of the play with one compound line, let us observe that the event “we will have exactly w winning numbers in at least one simple line of the system” is identical with “we will have exactly w winning numbers among the r of the initial combination,” where 0 ≤ w ≤ n . This identity of events holds true because the simple lines of the system represent all possible p-size combinations of numbers from the played r. Denoting this event by Aw,r , the probability P(Aw,r ) is given by the same general formula of the winning probability deduced in the chapter titled Probabilities of Winning with Simple Lines, in which parameter r will replace parameter p: CwCn−w P(A)=r m−r w,r Cmn .............. missing part ................... Adding together the three partial results, we obtain a winning probability at no lower than the third category of about 6.24%. Still related to the compound lines, we must make the following important observation: the unfolded system corresponding to a compound line does not satisfy the exclusiveness condition. This happens because, since it contains all possible p-size combinations of the r chosen numbers, in the system there will exist simple lines differing from one another by one single number, so they will have p – 1 common numbers. For this reason, the probability of a compound line winning in a specific category (eventually cumulated) will be lower than that of a system with an identical number of simple lines which adheres to the exclusiveness condition. There is also an advantage to the play with compound lines – namely, the number of possible prizes. We shall see in the next section that eventually winning in a certain category also implies winning in lower categories, and we will estimate rigorously the number of these prizes. 41PDF Image | THE MATHEMATICS OF LOTTERY Odds, Combinations, Systems
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