Probability of forming a triangle

winniethepooh wrote:Suppose you cut a stick in two random places. What is the probability that you'll be able to form a triangle out of the three pieces?

-----x--------------M--------y-----------

Let us assume that M is the midpoint of the stick and x and y are the points where the cuts have been made.

Now a triangle is possible if and only if no part is longer than half of the length of the stick.

Now if both cuts are on the same side of the midpoint M, then no triangle is possible. The probability that both cuts are on the same side of the midpoint M is 1/2.

Now there are two possible scenarios in case both the cuts x and y are on different sides of M,

• 1. -----x--------------M--------y-----------
  x is further left in its half than y is in the right half. In this case there is no triangle possible.
  
  2. -------------x------M--------y-----------
  x is not further left in its half than y is in the right half. In this case a triangle is possible.

Now (1) and (2) have equal probability equal to 1/2.

Hence, probability that a triangle is possible = (1/2)*(1/2) = 1/4