Question

If a stick is cut into two parts, find the expectation of maximum length of the two parts.

Answer 1

If the stick is broken at a random point which has equal probability at all the points on the stick, then the expected value of the length of the piece is 0.5.

However, there is a chance that the stick might break in a manner where one piece is longer than the other. So, if there is a larger piece, then it point of breaking is in between the middle point and the end point of the stick i.e. half the length of the stick + average of the remaining stick. Therefore, the maximum length of the larger piece could be 0.75.