Real Number Picking

Real number picking is the random selection of real numbers. Consider two independent random variables and , each with a uniform distribution on . What is the probability that , where denotes the nearest integer function, is even (Klosinski et al. 1994, p. 728)? Ratios exactly midway between integers have probability zero, so the tie-breaking convention does not affect the answer (Klosinski et al. 1994, p. 732).

The answer may be found as follows. For intervals lying on either side of 1, the relevant probabilities are

			(1)
		${int_0^1int_(ay)^(by)dxdy=1/2(b-a) for 0<=a<b<=1; int_0^1int_(x/b)^(x/a)dydx=1/(2a)-1/(2b) for 1<=a<b,$	(2)

(OEIS A091651).

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More things to try:

American Mathematics Competitions. Problem B3 in "William Lowell Putnam Competition Archive, 1993" https://web.archive.org/web/20010501141502/http://www.unl.edu/amc/a-activities/a7-problems/putnam/.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, p. 139, 2003.Klosinski, L. F.; Alexanderson, G. L.; and Larson, L. C. "The Fifty-Fourth William Lowell Putnam Mathematical Competition." Amer. Math. Monthly 101, 725-734, 1994. https://doi.org/10.1080/00029890.1994.11997019.Scholes, J. "54th Putnam 1993." Jan. 6, 2001. https://prase.cz/kalva/putnam/putn93.html.Sloane, N. J. A. Sequence A091651 in "The On-Line Encyclopedia of Integer Sequences."