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A242674
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Decimal expansion of the asymptotic probability of success in one of the Secretary problems.
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5
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5, 8, 0, 1, 6, 4, 2, 2, 3, 9, 2, 0, 8, 5, 5, 3, 4, 6, 4, 2, 6, 0, 0, 8, 3, 2, 3, 5, 7, 2, 9, 9, 7, 2, 7, 6, 6, 3, 3, 0, 8, 8, 6, 3, 8, 1, 1, 1, 1, 0, 1, 4, 0, 4, 3, 1, 6, 8, 7, 4, 1, 1, 7, 9, 2, 1, 6, 6, 1, 3, 8, 7, 7, 9, 6, 9, 2, 9, 2, 4, 9, 1, 8, 4, 5, 9, 3, 1, 5, 2, 6, 8, 4, 4, 7, 0, 3, 4, 7, 4
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OFFSET
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0,1
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COMMENTS
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This is the asymptotic probability of success for the full-information problem with uniform distribution: we can not only determine which of any two applicants is better than the other, but also determine his/her absolute value, and that value is known to be uniformly distributed on a known interval (say, [0, 1]), independently for each applicant; so we have more information than in the basic version of the problem (for which the chance of success is given by A068985), so the chance of success is greater. Here the number of applicants is known in advance (although we consider the limiting case when it is sent to infinity); for the variant where it is itself a random variable, see A325905. - Andrey Zabolotskiy, Sep 14 2019
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.15, p. 362.
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LINKS
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FORMULA
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exp(-a) - (exp(a)-a-1)*Ei(-a), where a is the unique real solution of the equation Ei(a)-gamma-log(a) = 1, Ei being the exponential integral function, and gamma the Euler-Mascheroni constant (0.5772156649...).
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EXAMPLE
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0.580164223920855346426008323572997276633...
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MATHEMATICA
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a = x /. FindRoot[ExpIntegralEi[x] - EulerGamma - Log[x] == 1, {x, 1}, WorkingPrecision -> 105]; Exp[-a] - (Exp[a]-a-1)*ExpIntegralEi[-a] // RealDigits[#, 10, 100]& // First
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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