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A075549 Decimal expansion of 9 - 12*log(2). 3

%I #28 May 20 2023 04:30:59

%S 6,8,2,2,3,3,8,3,3,2,8,0,6,5,6,2,8,6,9,9,3,2,1,4,5,4,2,5,0,1,8,8,1,1,

%T 8,3,0,9,3,9,9,8,3,8,7,6,7,6,9,3,6,9,5,0,5,5,1,8,3,9,8,8,6,0,7,9,2,7,

%U 6,5,3,6,3,6,3,6,6,3,4,1,2,7,2,9,6,4,0,0,7,6,0,4,2,9,7,5,7,4,9,4,9,5,9,8

%N Decimal expansion of 9 - 12*log(2).

%C Choose two numbers at random from the interval [0,1] (using a uniform distribution). This will give three subintervals of lengths a, b and c. What is the probability that there is a triangle with sides a, b and c? Given that a such a triangle exists, what is the probability that it is obtuse? Answer: Probability that there is a triangle is 1/4. Probability for this triangle to be obtuse is = 9 - 12 * log(2) = 0.68223... .

%C The problem proposed by Singmaster (1973) is to the calculate the probability that the three segments can form an obtuse triangle. Its solution is 1/4 of this constant, i.e., 9/4 - 3*log(2) = 0.170558... . - _Amiram Eldar_, May 20 2023

%D Paul J. Nahin, Digital Dice: Computational Solutions to Practical Probability Problems, Princeton University Press, 2008, pp. 31-32.

%H Zak Levi, <a href="http://faculty.missouristate.edu/l/lesreid/Adv28.html">The Problem #28</a>.

%H Robert Nelson, <a href="https://www.jstor.org/stable/3026739">Pictures, probability, and paradox</a>, The Two-Year College Mathematics Journal, Vol. 10, No. 3 (1979), pp. 182-190; <a href="https://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/00494925.di020687.02p00906.pdf">alternatove link</a>. See Problem 3, p. 184.

%H David Singmaster, <a href="http://www.jstor.org/stable/2688582">Problem 858</a>, Mathematics Magazine, Vol. 46, No. 1 (1973), p. 43; <a href="http://www.jstor.org/stable/2688271">Probability of an Obtuse Triangle</a>, Solution to Problem 85 by Major G. C. Holterman, ibid., Vol. 46, No. 5 (1973), pp. 294-295.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%e 0.682233833280656286993214542501881183...

%t RealDigits[9 - 12Log[2], 10, 100][[1]] (* _Alonso del Arte_, Nov 02 2013 *)

%o (PARI) 9 - 12*log(2) \\ _Michel Marcus_, Oct 14 2020

%Y Cf. A002162 (log(2)).

%K nonn,cons

%O 0,1

%A _Zak Seidov_, Oct 11 2002

%E Offset corrected by _R. J. Mathar_, Feb 05 2009

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