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2017 AMC 12B Problem 20

Problem 20 of 25HarderAlgebraProbability & Statistics

Real numbers xx and yy are chosen independently and uniformly at random from the interval (0,1).(0, 1). What is the probability that ⌊log⁡2x⌋=⌊log⁡2y⌋,\lfloor \log_2 x \rfloor = \lfloor \log_2 y \rfloor, where ⌊r⌋\lfloor r \rfloor denotes the greatest integer less than or equal to the real number r?r?

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Solution

For each positive integer n,n, ⌊log⁡2x⌋=−n\lfloor \log_2 x \rfloor = -n exactly when 12n≤x<12n−1,\dfrac{1}{2^n} \le x \lt \dfrac{1}{2^{n-1}}, an interval of length 12n.\dfrac{1}{2^n}. The event that both floors equal −n-n is a square of area 14n.\dfrac{1}{4^n}. Summing over all n,n, the probability is ∑n=1∞14n=141−14=13.\sum_{n=1}^{\infty} \frac{1}{4^n} = \frac{\frac{1}{4}}{1 - \frac{1}{4}} = \frac{1}{3}. Thus, the correct answer is D.
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Tagged: geometric probability · floor and ceiling functions · geometric sequence

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