You break a stick at two random points. What is the probability the three pieces form a triangle?

You break a stick at two random points. What is the probability the three pieces form a triangle?

Approach: Use triangle inequality constraints. Represent breaking points as x,y in [0,1] with x<y.

1/4. The conditions are: x < 0.5, y > 0.5, y-x < 0.5. Area satisfying these in unit square is 1/4.

Follow-up: What is the expected area of the triangle formed?

Key concepts: triangle inequality, uniform breaking, area method.