Problem Definition

I have a bar of chocolate whose length is L. If I randomly split it into two pieces, can you tell me the expected length of the larger piece? An equivalent question is, what is the resulting length of the larger piece on average?

Note: when I split the chocolate, I always cut the L-length edge, and the cut is parallel to the other edge.

Solution

Essentially, this problem asks us to compute the expectation of a random variable, which is the length of the larger piece. Let’s represent this random variable as letter $X$. The implication of “I randomly split it into two pieces” is that X satisfies uniform distribution $U(L/2, L)$.

Since the expectation of $U(a, b)$ is $(a+b)/2$, the expected length of the larger piece of chocolate is $3L/4$.