Bilkent University Fall 2022
MATH 255 - Probability and Statistics
Solutions to Midterm Exam II
Problem 1. [10pt] A stick is broken into three pieces by picking two points independently and
uniformly along the stick, and breaking the stick at those two points. What is the probability
that the three pieces can be assembled into a triangle?
Solution: Consider the case where x > y without loss of generality. Then, the segments have
lengths: y, x − y, and 1 − x. To form a triangle, they must satisfy the triangle inequality that
(+3pt)
(x − y) + y > 1 − x,
y + (1 − x) > x − y,
(x − y) + (1 − x) > y.
This yields that if x > 1/2, y < 1/2 and x − y < 1/2, then the segments form a triangle
conditioned on x > y (+3pt). The following figure illustrates the event of interest and the
conditioned event.
Hence, the conditional probability is 1/4 (+4pt).
1 09-25-2025 13:39:15 GMT -05:00
This study source was downloaded by 100000899606396 from CourseHero.com on
https://www.coursehero.com/file/195361566/MT2-solutions-1pdf/
MATH 255 - Probability and Statistics
Solutions to Midterm Exam II
Problem 1. [10pt] A stick is broken into three pieces by picking two points independently and
uniformly along the stick, and breaking the stick at those two points. What is the probability
that the three pieces can be assembled into a triangle?
Solution: Consider the case where x > y without loss of generality. Then, the segments have
lengths: y, x − y, and 1 − x. To form a triangle, they must satisfy the triangle inequality that
(+3pt)
(x − y) + y > 1 − x,
y + (1 − x) > x − y,
(x − y) + (1 − x) > y.
This yields that if x > 1/2, y < 1/2 and x − y < 1/2, then the segments form a triangle
conditioned on x > y (+3pt). The following figure illustrates the event of interest and the
conditioned event.
Hence, the conditional probability is 1/4 (+4pt).
1 09-25-2025 13:39:15 GMT -05:00
This study source was downloaded by 100000899606396 from CourseHero.com on
https://www.coursehero.com/file/195361566/MT2-solutions-1pdf/
