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}
.