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097Bertrand's Paradox: The Random Chord ProblemConsider a circle of radius r with an inscribed equilateral triangle. A chord is drawn 'at random.' What is the probability that the chord is longer than the side of the triangle? Compute the answer under each of the following three methods of selecting a random chord: (a) **Random endpoints:** Fix one endpoint on the circle and choose the other endpoint uniformly on the circumference. (b) **Random midpoint:** Choose the midpoint of the chord uniformly in the disk. (c) **Random radius:** Choose a radius, then choose a point uniformly along that radius as the midpoint of the chord. For each method, set up and evaluate the relevant integral or geometric argument.概率简单derivation未尝试免费098The Will Rogers Paradox (Stage Migration)Group A consists of the values \ 1, 2, 3, 9\ and Group B consists of \ 5, 6, 7, 8\ . (a) Compute the mean of each group. (b) Now move the element 5 from Group B to Group A. Recompute the mean of each group. Verify that both means have increased. (c) The overall mean across all 8 values is unchanged. Prove the following general statement: if an element is moved from group B to group A, and the element's value x satisfies A < x < B (where A , B are the pre-move group means), then both post-move group means strictly exceed their pre-move values. (d) A medical study reports that a new diagnostic technology has improved average survival time in both Stage I and Stage II cancer patients, yet overall survival is unchanged. Explain how this is possible without any actual improvement in treatment.概率中等数值题未尝试免费