Given $n \in \mathbb{N}$ consider the set of all $n \times n$ matrices with cells whose values are in the range $\left[0,1\right]$, define this set as $\mathcal{M}$ . Define a probability distribution over $\mathcal{M}$ in the following way: $$ \quad \mathbb{P}\left(\forall i,j \in \left[n\right]\quad a_{i,j} \leq M_{i,j} \leq b_{i,j} \right) = \prod_{i,j}\left(b_{i,j}-a_{i,j}\right) $$ (i.e. the probability distribution over $\mathcal{M}$ was created by allowing each cell in the matrix to vary independently uniformly over $\left[0,1\right]$).
Now consider the determinant (det) as a random variable over $\mathcal{M}$ with the probability distribution we constructed. What is the distribution of det?
for $n=1$ the solution is obvious, but for bigger $n$ I believe it might be the Laplace distribution of some form but I'm not sure how to prove it.