Answer to Question #225802 in Statistics and Probability for Ansi

Question #225802
A rope, 12 meter long, is cut into two pieces at a random point between its ends. What
is the probability that the length of one piece is at least thrice the length of the other?
1
Expert's answer
2021-08-15T18:04:03-0400

Let X=X= the length of the shorter piece of rope, so that X6 m.X\leq 6\ m.

The random variable XX is uniformly distributed on the interval 0X6,0\leq X\leq 6, so


P(0Xx)=x6P(0\leq X\leq x)=\dfrac{x}{6}

If X12X13,\dfrac{X}{12-X}\leq\dfrac{1}{3}, so


12X3X12-X\geq3X

X3X\leq3

P(0X3)=36=12P(0\leq X\leq 3)=\dfrac{3}{6}=\dfrac{1}{2}

The probability that the length of one piece is at least thrice the length of the other is 12.\dfrac{1}{2}.



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