A system containing $n$ qubits is described by a $2^n-$dimensional Hilbert space. Some of these states can be decomposed as product states, but not all of them. The remaining ones are called entangled states. In fact, let
$$ |\Psi\rangle = \sum_{k=1}^{2^n} c_k |\phi_k\rangle \tag{1}$$
be a general state in this Hilbert space, where $c_k\in \mathbb{C}$ and $\langle \phi_j|\phi_k\rangle =\delta_{jk}$ are the complex coefficients and orthonormal basis states. A product state in this Hilbert space is written as
$$|\Psi'\rangle = \begin{pmatrix} a_1 \\ b_1 \end{pmatrix} \otimes \begin{pmatrix} a_2 \\ b_2 \end{pmatrix} \otimes \cdots \otimes \begin{pmatrix} a_n \\ b_n \end{pmatrix}. \tag{2}$$
It's only when the $c_k$ in Eq. $(1)$ match specific conditions that $|\Psi\rangle$ can be written in the form of Eq. $(2)$. I could be wrong, but the condition that I arrived at was
$$\frac{c_1}{c_2}=\frac{c_3}{c_4}=\cdots=\frac{c_{k-1}}{c_k}=\cdots=\frac{c_{2^n-1}}{c_{2^n}}, \tag{3}$$
where $a_i,b_i \in \mathbb{C}$. The "volume" occupied by states that satisfy these conditions and are therefore product states is of course smaller than the one occupied by the ones that don't. My question is: how much smaller? Is there a way to measure how more frequent the entangled states are as compared to the product states using the constraint in Eq. $(3)$?
