By a profound result of Heinrich, Novak, Wasilkowski, and Woźniakowski the inverse of the star-discrepancy
n
∗
(
s
,
ε
)
n^*(s,\varepsilon )
satisfies the upper bound
n
∗
(
s
,
ε
)
≤
c
a
b
s
s
ε
−
2
n^*(s,\varepsilon ) \leq c_{\mathrm {abs}} s \varepsilon ^{-2}
. This is equivalent to the fact that for any
N
N
and
s
s
there exists a set of
N
N
points in
[
0
,
1
]
s
[0,1]^s
whose star-discrepancy is bounded by
c
a
b
s
s
1
/
2
N
−
1
/
2
c_{\mathrm {abs}} s^{1/2} N^{-1/2}
. The proof is based on the observation that a random point set satisfies the desired discrepancy bound with positive probability. In the present paper we prove an applied version of this result, making it applicable for computational purposes: for any given number
q
∈
(
0
,
1
)
q \in (0,1)
there exists an (explicitly stated) number
c
(
q
)
c(q)
such that the star-discrepancy of a random set of
N
N
points in
[
0
,
1
]
s
[0,1]^s
is bounded by
c
(
q
)
s
1
/
2
N
−
1
/
2
c(q) s^{1/2} N^{-1/2}
with probability at least
q
q
, uniformly in
N
N
and
s
s
.