An m × n m \times n matrix E E with n n ones and ( m − 1 ) n (m - 1)n zeros, which satisfies the Pólya condition, may be regular and singular for Birkhoff interpolation. We prove that for random distributed ones, E E is singular with probability that converges to one if m m , n → ∞ n \to \infty . Previously, this was known only if m ⩾ ( 1 + δ ) n / log ⁡ n m \geqslant (1 + \delta )n/\log n . For constant m m and n → ∞ n \to \infty , the probability is asymptotically at least 1 2 \tfrac {1} {2} .