I came to know this problem from Dr. W. Bryc's slides (at University of Cincinnati), and I have been continually working on this problem for almost 5 days using different techniques. But I am only able to understand the geometry and algebra for the problem when $n=3$.

**Background and Physical Meaning of Problem**

This research problem was proposed by late Samule Karlin, who knew the answer but has never published it. As I realized yesterday, it can be understood as the probability of $n$ independent random walks, with positively weighted uniform step size, all starting at $0$ but all reaching $1$ at step $n$. The solution itself seems to have strong symmetry but this symmetry seems to be hard to exploit due to dependence.

**Satement of Problem**

The problem and solution, both due to Samuel Karlin', are stated below:

Let the $n \times n$ matrix $\mathbf{U}=(u_{ij})$ have entries that are independent and identically distributed on the unit interval $[0,1]$, and let $\mathbb{R}_{+}^{n} = \{\mathbf{y}=(y_1,\ldots,y_n) \in \mathbb{R}^n: y_i >0 \text{ for } i=1,\ldots,n\} $, where $\mathbb{R}^n$ is the $n$-dimensional Euclidean space. Then \begin{equation} \Pr(\mathbf{U}: \mathbf{U}\mathbf{x}=\mathbf{1} \text{ for some } \mathbf{x} \in \mathbb{R}_{+}^{n}) = \dfrac{1}{2^{n-1}}, \end{equation}

where $\Pr$ is the probability measure with respect to $\mathbf{U}$, $\mathbf{1}$ is a vector of $n$ $1$'s, and all vectors are column vectors.

**Latest update**

I spent sometime on this tough problem during the vacation and found out the following when $n \geq 3$: (a) the distribution of $\pi(c_i)$ is NOT symmetric with respect to (wrt) $0$ because its characteristic function is not a real-valued function, where $c_i$ is the $i$th column of $\mathbf{U}$; this means that the distribution of $\eta(\pi(c_i))$ is NOT symmetric wrt to $0$; (b) Wendel's argument (Wendel 1962, a problem in geometric probability) can not be directly applied to $\eta(\pi(c_i))$ due to (a) and because the vectors $\eta(\pi(c_i))$ are restricted to be in the orthogonal complement of $\mathbf{1}$; (c) however, Greg's strategy definitely will help resolve the problem. For meaning of these notations, please see comment by Greg.

**Attempts Made**

**Conditioning**: splitting $\mathbf{U}$ into a $(n-1) \times (n-1)$ principal submatrix $\mathbf{W}$ and $\mathbf{x}$ into an $(n-1)$-dimensional subvector and the $n$th entry. But such conditioning does not seem to help since the induced joint equations for the solution in $\mathbf{x}$ involves the volume form with respect to the inverse $\mathbf{W}^{-1}$.**Geometry**: for $n=3$ the problem has a clear geometric meaning, in that two random planes cut the unit cube, and the relative positions of the columns of $\mathbf{U}$ and $\mathbf{1}$ have to remain in a configuration in order for $\mathbf{x}$ to exist. However, when $n>4$, we can not visualize the configuration that induces the solution $\mathbf{x}$.

**Any comments/suggestions on how to derive Karlin's solution? Thank you.**