Harder probability round for research roles.
Breaking a stick into a triangle
The question
A stick of length one is broken at two points chosen uniformly and independently. What is the probability the three pieces form a triangle?
Have a go before you open the solution. If you are stuck after a few minutes, the first paragraph below is usually enough of a nudge.
Show the worked solution
The answer
1/4.
Let the break points be x and y, uniform on the unit square. The pieces form a triangle when no piece exceeds 1/2.
Take x < y. The pieces are x, y - x and 1 - y. All three are under 1/2 when x < 1/2, y > 1/2 and y - x < 1/2.
That region is a triangle of area 1/8 inside the half-square of area 1/2. By symmetry the case y < x contributes another 1/8.
Total favourable area is 1/4, so the probability is 1/4. Drawing the unit square is the fastest route, and interviewers expect to see the picture.
What it is testing
Turning a worded constraint into a region and computing its area.
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