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Mathematical Statistics: Probability for Length of a Stick

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A point is chosen uniformly over a 1-yard wooden stick, and a mark is made. The procedure is repeated, independently, and another mark is made. The stick is then sawn at the two marks, yielding three shorter sticks. What is the probability that at least one of those sticks is at least ½ yard long?

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Step-by-steps explanations and computations are given to find the probability of the length of a stick. The file includes 4 figures.

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Since the points are marked independently of each other, the points are uniformly distributed on (0,1). Consider the 1-yard wooden stick, measured starting from the left side, with a cut at x and a cut at y. Note that x can be to the left of y, or x can be to the right of y.

There are 3 mutually exclusive cases whereby a stick of length ½ yard or greater is created
Case 1 : Both cuts are to the left of the ½-way point on the 1-yard stick. This means that there is a stick of length greater than ½ to the right of y. An example is shown below.

0 x y ½ ...

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  • MSc, California State Polytechnic University, Pomona
  • MBA, University of California, Riverside
  • BSc, California State Polytechnic University, Pomona
  • BSc, California State Polytechnic University, Pomona
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