HMMT 二月 2014 · 冲刺赛 · 第 36 题

英文解析

36. [ 25 ] We have two concentric circles C and C with radii 1 and 2, respectively. A random chord of C
    1 2 2
    is chosen. What is the probability that it intersects C ?
    m 1
    Your answer to this problem must be expressed in the form , where m and n are positive integers. Ifn
    25 · Xyour answer is in this form, your score for this problem will be b c , where X is the total number ofteams who submit the answer (including your own team), and Y is the total number of teams whom ynsubmit a valid answer. Otherwise, your score is 0. (Your answer is not graded based on correctness,
    whether your fraction is in lowest terms, whether it is at most 1, etc.)
    Answer: N/A The question given at the beginning of the problem statement is a famous problemin probability theory widely known as Bertrand’s paradox. Depending on the interpretation of thephrase “random chord,” there are at least three different possible answers to this question:
    • If the random chord is chosen by choosing two (uniform) random endpoints on circle C andtaking the chord joining them, the answer to the question is 1 / 3.2
    • If the random chord is chosen by choosing a (uniformly) random point P the interior of C (otherthan the center) and taking the chord with midpoint P , the answer to the question becomes 1 / 4.2
    • If the random chord is chosen by choosing a (uniformly) random diameter d of C , choosing apoint P on d , and taking the chord passing through P and perpendicular to d , the answer tothe question becomes 1 / 2. (This is also the answer resulting from taking a uniformly randomhorizontal chord of C .)
    You can read more about Bertrand’s paradox online at http://en.wikipedia.org/wiki/Bertrand_2
    paradox_(probability) . We expect that many of the valid submissions to this problem will be equalto 1 / 2 , 1 / 3, or 1 / 4.
    However, your score on this problem is not based on correctness, but is rather proportional to the number of teams who wrote the same answer as you! Thus, this becomes a problem of finding whatis known in game theory as the “focal point,” or “Schelling point.” You can read more about focalpoints at http://en.wikipedia.org/wiki/Focal_point_(game_theory) or in economist Thomas
    Schelling’s book The Strategy Of Conflict .