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2966Longer Piece Above Two ThirdsA unit stick is cut once at a uniformly random point. What is the probability that the longer piece has length more than 2/3?概率简单derivation未尝试面试订阅2967Both Pieces Longer Than One FifthA unit stick is cut once at a uniformly random point. What is the probability that both resulting pieces have length greater than 1/5?概率简单derivation未尝试面试订阅2968Expected Shorter Piece from One CutA unit stick is cut once at a uniformly random point. Let S be the length of the shorter piece. Compute E[S].概率简单derivation未尝试面试订阅2969Expected Product of the Two PiecesA unit stick is cut once at a uniformly random point. If the two piece lengths are U and 1-U, compute E[U(1-U)].概率简单derivation未尝试面试订阅2970Expected Absolute Imbalance After One CutA unit stick is cut once at a uniformly random point. Let D be the absolute difference between the two resulting lengths. Compute E[D].概率简单derivation未尝试面试订阅2971Three Pieces All Longer Than One FifthTwo cut points are drawn independently and uniformly on [0,1], and the stick is cut at the ordered locations. What is the probability that all three resulting pieces have length greater than 1 5 ?概率中等derivation未尝试面试订阅2973Triangle Condition for a Broken StickTwo cut points are drawn independently and uniformly on [0,1], and the stick is cut into three pieces. What is the probability that the three pieces can form a triangle?概率中等derivation未尝试面试订阅2974Longest Piece Shorter Than Two ThirdsTwo cut points are drawn independently and uniformly on [0,1]. After ordering them, the stick is split into three pieces. What is the probability that the longest piece has length less than 2/3?概率中等derivation未尝试面试订阅2975Expected Shortest Piece with Two CutsTwo cut points are drawn independently and uniformly on [0,1], then ordered to form three stick pieces with lengths X 1,X 2,X 3. Compute E[\min(X 1,X 2,X 3)].概率中等derivation未尝试面试订阅2976Expected Longest Piece with Two CutsTwo cut points are drawn independently and uniformly on [0,1], then ordered to form three stick pieces with lengths X 1,X 2,X 3. Compute E[\max(X 1,X 2,X 3)].概率中等derivation未尝试面试订阅2977Expected Sum of Squared Piece LengthsTwo cut points are drawn independently and uniformly on [0,1], then ordered to form three stick pieces with lengths X 1,X 2,X 3. Compute E[X 1 2+X 2 2+X 3 2].概率中等derivation未尝试面试订阅2978Expected Product of the Three PiecesTwo cut points are drawn independently and uniformly on [0,1], then ordered to form three stick pieces with lengths X 1,X 2,X 3. Compute E[X 1X 2X 3].概率中等derivation未尝试面试订阅2979Leftmost Piece Exceeds One HalfTwo cut points are drawn independently and uniformly on [0,1] and ordered. Let L be the leftmost piece length. What is P(L>1/2)?概率中等derivation未尝试面试订阅2980Expected Leftmost Piece LengthTwo cut points are drawn independently and uniformly on [0,1] and ordered. Let L be the leftmost piece length. Compute E[L].概率中等derivation未尝试面试订阅2981Size-Biased Piece from Two CutsTwo cut points are drawn independently and uniformly on [0,1], forming three pieces with lengths X 1,X 2,X 3. A random point is then drawn uniformly on the stick, and you keep the piece containing that point. Let B be its length. Compute E[B].概率困难derivation未尝试面试订阅2982Second Moment of a Size-Biased PieceIn the same setup as above, a random point is drawn on the stick after two cuts and B is the length of the piece containing that point. Compute E[B 2].概率困难derivation未尝试面试订阅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