Let X= the length of the shorter piece of rope, so that X≤6 m.
The random variable X is uniformly distributed on the interval 0≤X≤6, so
P(0≤X≤x)=6xIf 12−XX≤31, so
12−X≥3X
X≤3
P(0≤X≤3)=63=21The probability that the length of one piece is at least thrice the length of the other is 21.
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