Interview Question for Quantitative Compensation Analyst at Google:

Its 1/4.

Here is the key idea for my analysis of the problem:

  If we consider the original stick to be of unit length, then
  we can form a triangle whenever the longest stick is less than
  a half unit long.

Suppose x is the length of the first piece and y is the length of the second piece (both must be nonnegative). Then y will be <= 1-x, and to be able to form a triangle, y must be <= .5 - x with x <= .5. The probability of being able to form a triangle is the area of the second set of (x,y) pairs divided by the area of the first set of (x,y) pairs, which is .125/.5 = .25.

The probability is 0 given that it is a question about the probability of 2 breaking points falling on 1st thirdth and 2nd thirdth point. For any continuous variable, the probability of the variable equal to countable points (including indefinite countable numbers) equal to 0