A ruler of length 2m is broken at a point selected with uniform density along its length. The left part is discarded and
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A stick of length 22 m is made of uniformly dense material. A point is chosen randomly on the stick and the stick is broken at that point. The left portion of the stick is discarded and now again another point is chosen randomly on the remaining right portion of the stick and the stick is broken again at that point and the left part is again discarded.The process is continued indefinitely.What is the probability that one of the discarded left parts has length >1>1 m?
Formulating this problem we basically have a sequence of random variables {XnXn} where X1∼U(0,2)X1∼U(0,2) , X2|X1∼U(0,2−X1)X2|X1∼U(0,2−X1),X3|X1,X2∼U(0,2−X1−X2)X3|X1,X2∼U(0,2−X1−X2) and so on. The probability that any one of the discarded parts is more than 11 m is equivalent to say that it is 1−P(∩1−P(∩{Xi<1Xi<1}) But I cannot find the probability explicitly as it is dependent on X1X1.