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2941Two Random Times More Than 20 Seconds ApartTwo independent timestamps are drawn uniformly from the interval [0,60] seconds. What is the probability that they differ by more than 20 seconds?概率中等derivation未尝试面试订阅2942Expected Gap Between Two Random Points on an IntervalIf X and Y are independent Uniform (0,1) points, compute E[|X-Y|].概率中等derivation未尝试面试订阅2943Two Random Points Within 0.1 of Each OtherTwo independent points are drawn uniformly from [0,1]. What is the probability that their distance is less than 0.1?概率简单derivation未尝试面试订阅2944Midpoint Beyond Two ThirdsIf X and Y are independent Uniform (0,1) points, what is the probability that their midpoint exceeds 2/3?概率简单derivation未尝试面试订阅2945Coordinatewise Comparability in the Unit SquareTwo independent random points (X 1,Y 1) and (X 2,Y 2) are drawn uniformly from the unit square. What is the probability that one point is coordinatewise less than the other, i.e. either X 1<X 2 and Y 1<Y 2, or X 1>X 2 and Y 1>Y 2?概率中等derivation未尝试面试订阅2946Midpoint Inside the Central SquareTwo independent random points are drawn uniformly from the unit square. What is the probability that their midpoint lies in the central square [1/4,3/4] [1/4,3/4]?概率中等derivation未尝试面试订阅2948Closer to Vertical Sides Than Horizontal SidesA random point is drawn uniformly from the unit square. What is the probability that its distance to the nearer vertical side is smaller than its distance to the nearer horizontal side?概率中等derivation未尝试面试订阅2949Expected Distance to the Nearest Vertical SideA random point (X,Y) is uniform on the unit square. Compute the expected distance from the point to the nearer of the two vertical sides.概率简单derivation未尝试面试订阅2950Radial CDF in the Unit DiskA random point is uniform in the unit disk. Let R be its distance from the center. Compute P(R\le r) for 0\le r\le 1.概率简单derivation未尝试面试订阅2951Expected Radius in the Unit DiskA random point is uniform in the unit disk. Let R be its distance from the center. Compute E[R].概率中等derivation未尝试面试订阅2952Expected Squared Radius in the Unit DiskA random point is uniform in the unit disk. Compute E[R 2], where R is its distance from the center.概率简单derivation未尝试面试订阅2953Expected Larger Radius from Two Disk PointsTwo independent random points are drawn uniformly from the unit disk. Let R 1 and R 2 be their distances from the center, and let \[ M=\max(R 1,R 2). \] Compute E[M].概率中等derivation未尝试面试订阅2954A Sector of Angle Pi over SixA random point is uniform in the unit disk. What is the probability that its polar angle lies in a fixed sector of angle /6?概率简单derivation未尝试面试订阅2955Inner Similar Triangle in the Right TriangleA random point is uniform in the triangle \[ T=\ (x,y): x\ge 0,\ y\ge 0,\ x+y\le 1\ . \] What is the probability that the point also satisfies x+y\le 1/2?概率简单derivation未尝试面试订阅2956Expected x-Coordinate in the Right TriangleA random point is uniform in \[ T=\ (x,y): x\ge 0,\ y\ge 0,\ x+y\le 1\ . \] Compute E[X].概率中等derivation未尝试面试订阅2957Closer to the x-Axis or to the y-Axis?A random point is uniform in the triangle \[ \ (x,y): x\ge 0,\ y\ge 0,\ x+y\le 1\ . \] What is the probability that the point is closer to the x-axis than to the y-axis?概率简单derivation未尝试面试订阅2958The Medial TriangleA random point is uniform in a triangle. What is the probability that it lies inside the medial triangle formed by joining the midpoints of the three sides?概率简单derivation未尝试面试订阅2959Shorter Arc Less Than Pi over ThreeTwo independent random points are chosen uniformly on the unit circle. What is the probability that the shorter arc between them has angular length less than /3?概率中等derivation未尝试面试订阅2960Expected Shorter Arc Between Two Circle PointsTwo independent random points are chosen uniformly on the unit circle. Let \Theta be the angular length of the shorter arc between them. Compute E[\Theta].概率中等derivation未尝试面试订阅2961Coordinate Differences Both Below One HalfTwo independent random points are drawn uniformly from the unit square. What is the probability that both coordinate differences satisfy \[ |X 1-X 2|<\frac12,\qquad |Y 1-Y 2|<\frac12? \]概率中等derivation未尝试面试订阅