cp's OEIS Frontend

A242674 Decimal expansion of the asymptotic probability of success in one of the Secretary problems.

%I A242674 #15 Feb 16 2025 08:33:22
%S A242674 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,
%T A242674 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,
%U A242674 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
%N A242674 Decimal expansion of the asymptotic probability of success in one of the Secretary problems.
%C A242674 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
%D A242674 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.15, p. 362.
%H A242674 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/SultansDowryProblem.html">Sultan's Dowry Problem</a>
%H A242674 Wikipedia, <a href="http://en.wikipedia.org/wiki/Secretary_problem">Secretary problem</a>
%F A242674 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...).
%e A242674 0.580164223920855346426008323572997276633...
%t A242674 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
%Y A242674 Cf. A054404, A242672, A242673, A068985, A325905.
%K A242674 nonn,cons
%O A242674 0,1
%A A242674 _Jean-François Alcover_, May 20 2014