Take a stick of unit length and break it into three pieces c

Take a stick of unit length and break it into three pieces, choosing the break points at random. (The break points are assumed to be chosen simultaneously.) What is the probability that the three pieces can be used to form a triangle? Hint: The sum of the lengths of any two pieces must exceed the length of the third, so each piece must have length < 1/2.

Solution

The stick is broken into 3 pieces.

As sum of two pieces should not exceed the third we have

each piece <1/2

Prob that 3 pieces form a triangle = Prob (each piece <1/2)

= 1-P(one piece >1/2)

Any piece greater than 1/2 has equal prob as less than 1/2

Hence reqd prob = 0.5