A different approach to a common questionA unit stick is randomly broken into 3 pieces,...
A different approach to a common question
A unit stick is randomly broken into 3 pieces, it is given that these three pieces can make a triangle, what is the expected length of the medium-sized piece?
This a question we are all familiar with and anyone who has seen it anywhere/solved it knows that the expected length of the medium piece is , we reach this conclusion by using , and we know we can calculate E(L) and E(S), so we just use to get our answer, is there any way we can solely calculate the E(M) without calculating the other two values?
L: Length of the longest part
S: Length of the smallest part
M: Length of the medium part
Answer & Explanation
An easy approach.
Let's arrange the segments from smallest to largest. Let the three segments be x, and .
Now sum of all segments ;
with the conditions: and .
for normalisation, let n be a quantity , then,
x can be chosen randomly among a pool from [0,2n], y be chosen randomly from a pool [0,3n] and z from a pool [0,6n] where (x,y,z,n) R.
Therefore we can say:
Expected value of , expected value of and expected value of .
Expected length of middle segment =
Expected total length =
The the expected length of middle segment is