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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2813Largest Barycentric Piece Exceeds HalfA point is chosen uniformly inside a triangle. Joining the point to the three vertices partitions the triangle into three smaller triangles. What is the probability that the largest of those three area fractions exceeds 1/2?脑筋急转弯中等derivation未尝试面试订阅2814Closer to the Center Than to the OriginA point is chosen uniformly in the unit square. What is the probability that it is closer to the center (1/2,1/2) than to the origin (0,0)?脑筋急转弯中等derivation未尝试面试订阅5666Balanced Parenthesis CountHow many ways can you arrange 5 left and 5 right parentheses in a row so that every prefix has at least as many left as right parentheses (i.e. the sequence is balanced)?脑筋急转弯中等数值题未尝试免费5667Hat-Check Derangement CountSix guests check distinct hats. The attendant returns the hats in a random order. In how many of the possible return orderings does no guest receive their own hat?脑筋急转弯中等数值题未尝试免费5668Secret-Santa No Self-Draw ProbabilityFive people draw names uniformly at random from a hat containing all five of their own names, one draw each, forming a random permutation. What is the probability that nobody draws their own name?脑筋急转弯中等数值题未尝试免费5669Five Gifts Four Children SurjectionFive distinct gifts are distributed to 4 distinct children so that every child receives at least one gift. In how many ways can this be done?脑筋急转弯中等数值题未尝试免费