Question

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?

Answer 1

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

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

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

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

$$12-X\geq3X$$

$$X\leq3$$

$$P(0\leq X\leq 3)=\dfrac{3}{6}=\dfrac{1}{2}$$

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