Triple Sticks
Given a straight stick, person and person each choose a point on it independently at random. The stick is then broken exactly at these points into 3 pieces.
What is the probability that you can form a triangle with the 3 smaller sticks?
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1 solution
If we call the longest stick (or equal longest stick) created, c, and the length of the original stick, x, we know c must be at least 1/3x as otherwise another stick would have to be longer. At the same time the length of the two other sticks added together must be more than c as otherwise a triangle could not be formed.
The length of c can therefore be at most 1/2. Therefore, c, to form a triangle, is between 1/3x and 1/2x but could be between 1/3x and 1x. The probability of being able to form a triangle is: (1/2-1/3)/(1-1/3)=(1/6)/(2/3)= 1/4