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2983Four Pieces All Longer Than One SixthThree cut points are drawn independently and uniformly on [0,1] and ordered, producing four pieces. What is the probability that all four pieces are longer than 1/6?概率中等derivation未尝试面试订阅2984Longest Piece Below One Half with Three CutsThree cut points are drawn independently and uniformly on [0,1] and ordered, producing four pieces. What is the probability that the longest piece has length less than 1/2?概率中等derivation未尝试面试订阅2985Expected Shortest Piece with Three CutsThree cut points are drawn independently and uniformly on [0,1] and ordered, producing four pieces. Let M be the shortest piece length. Compute E[M].概率中等derivation未尝试面试订阅2986Expected Longest Piece with Three CutsThree cut points are drawn independently and uniformly on [0,1] and ordered, producing four pieces. Let L be the longest piece length. Compute E[L].概率困难derivation未尝试面试订阅2987Expected Sum of Squared Lengths with Three CutsThree cut points are drawn independently and uniformly on [0,1] and ordered, producing four pieces with lengths X 1,\dots,X 4. Compute E[X 1 2+\cdots+X 4 2].概率中等derivation未尝试面试订阅2988Expected Length of a Size-Biased Piece with Three CutsThree cut points are drawn independently and uniformly on [0,1]. After ordering them, a random point is chosen uniformly on the stick and you keep the piece containing that point. Let B be that piece length. Compute E[B].概率困难derivation未尝试面试订阅2989Expected Shortest Piece with Five PiecesFour cut points are drawn independently and uniformly on [0,1] and ordered, producing five pieces. Let M be the shortest piece length. Compute E[M].概率困难derivation未尝试面试订阅2990Expected Longest Piece with Five PiecesFour cut points are drawn independently and uniformly on [0,1] and ordered, producing five pieces. Let L be the longest piece length. Compute E[L].概率困难derivation未尝试面试订阅