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Winning probabilities In the compound and bridgehead lines we obtained, for the winning probabilities, general formulas whose variables were only the parameters that define the respective systems. This is not possible for reduced systems because not only the parameters, but also the particular structure of each system influences the probability of a certain winning event. Thus, we cannot obtain a general formula on the basis of which to calculate the winning probabilities for a given reduced system. Depending on the removal criterion, this calculation can theoretically be made with the help of a computer, on the base of some complex mathematical algorithms. The only situation in which a direct calculation is possible is that in which the reduced system satisfies the exclusiveness condition with respect to the winning event to be measured. In this case, the winning probability is linear with the number of simple lines in the system. But this situation is very rare for the most common lottery matrices and for their winning thresholds. Next we shall find a convenient approximation of the winning probability for any reduced system, which can replace a very laborious exact calculation. .............. missing part ................... Examples of approximations of the winning probability: 1) In the 6/49 matrix, we play a reduced system with 30 simple lines, coming from a 10-number compound line. Let us approximate the probability of winning with exactly 4 numbers. a. r'=10 ands=30. b. In table 6/49 from the section devoted to the probabilities of the compoundlines,forw=4and r'=10 wefind P(A'4)=0.011128. c. In the same table, in the second column, the number closest to 30andlessthanorequalto30is s''=28. d. At the intersection of the row corresponding to s '' = 28 with column w = 4 we find P(A''4 ) = 0.004105. e. We have then: 0.004105 < P(A4 ) < 0.11128. 56PDF Image | THE MATHEMATICS OF LOTTERY Odds, Combinations, Systems
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