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2791Average Radius in the Unit DiskA point is drawn uniformly from the unit disk. What is the expected distance from the point to the center?脑筋急转弯中等derivation未尝试面试订阅2792Right-Hand Circular Segment ProbabilityA point is drawn uniformly from the unit disk. What is the probability that its x-coordinate exceeds 1/2?脑筋急转弯困难derivation未尝试面试订阅2793Expected Chord Length on the Unit CircleTwo points are chosen independently and uniformly on the unit circle. What is the expected length of the chord connecting them?脑筋急转弯中等derivation未尝试面试订阅2794When a Midpoint-Generated Chord Exceeds the RadiusA chord of the unit circle is generated by choosing its midpoint uniformly in the disk. What is the probability that the chord length exceeds 1, the circle's radius?脑筋急转弯中等derivation未尝试面试订阅2795Four Points in a Common SemicircleFour points are chosen independently and uniformly on the unit circle. What is the probability that all four lie inside some semicircle?脑筋急转弯中等derivation未尝试面试订阅2797Expected Squared Distance in the Unit DiskTwo points are chosen independently and uniformly from the unit disk. What is the expected squared Euclidean distance between them?脑筋急转弯中等derivation未尝试面试订阅2798Average Manhattan Distance in the Unit SquareTwo points are chosen independently and uniformly in the unit square. What is their expected Manhattan distance |X 1-X 2|+|Y 1-Y 2|?脑筋急转弯中等derivation未尝试面试订阅2799A Unit-Interval Distance WindowIf U and V are independent Uniform (0,1) variables, what is the probability that |U-V|\le 1/4? Solve it geometrically.脑筋急转弯中等derivation未尝试面试订阅2800Average Distance to a Side of an Equilateral TriangleA point is chosen uniformly inside an equilateral triangle of side length 2. What is its expected distance to a fixed side?脑筋急转弯中等derivation未尝试面试订阅2801Staying Away From All Three SidesA point is chosen uniformly inside an equilateral triangle. What is the probability that its distance to each of the three sides is at least one-sixth of the altitude?脑筋急转弯中等derivation未尝试面试订阅2802Closer to the Diagonal Than to the Left EdgeA point is chosen uniformly in the unit square. What is the probability that it is closer to the diagonal line y=x than to the left edge x=0?脑筋急转弯困难derivation未尝试面试订阅2803Conditioning on the Outer Half of the DiskA point is chosen uniformly in the unit disk. Conditional on the point lying outside the disk of radius 1/2, what is its expected distance from the center?脑筋急转弯中等derivation未尝试面试订阅2804Chord Longer Than √2Fix one point on the unit circle and choose a second point uniformly on the circle. What is the probability that the chord length exceeds \sqrt2?脑筋急转弯中等derivation未尝试面试订阅2805Squared Distance to the Center of the SquareA point is chosen uniformly in the unit square. What is the expected squared distance from the point to the center (1/2,1/2)?脑筋急转弯中等derivation未尝试面试订阅2806Closer to the Center Than to the BoundaryA point is chosen uniformly in the unit disk. What is the probability that the point is closer to the center than to the boundary of the disk?脑筋急转弯中等derivation未尝试面试订阅2808Variance of a Coordinate in the DiskA point is chosen uniformly from the unit disk. Let X be its x-coordinate. Compute Var (X).脑筋急转弯中等derivation未尝试面试订阅2809Maximum Coordinate of Two Uniform NumbersTwo independent Uniform (0,1) variables U and V are drawn. What is E[\max(U,V)]? Give a geometric interpretation in the unit square.脑筋急转弯中等derivation未尝试面试订阅2810Expected Squared Chord LengthTwo points are chosen independently and uniformly on the unit circle. What is the expected squared length of the chord joining them?脑筋急转弯中等derivation未尝试面试订阅2811Expected Chord Length From a Uniform MidpointA chord of the unit circle is generated by choosing its midpoint uniformly in the disk. What is the expected chord length?脑筋急转弯中等derivation未尝试面试订阅2812Expected Squared Distance in the Unit SquareTwo points are chosen independently and uniformly in the unit square. What is the expected squared Euclidean distance between them?脑筋急转弯中等derivation未尝试面试订阅