Recently, Damgård, Landrock and Pomerance described a procedure in which a
k
k
-bit odd number is chosen at random and subjected to
t
t
random strong probable prime tests. If the chosen number passes all
t
t
tests, then the procedure will return that number; otherwise, another
k
k
-bit odd integer is selected and then tested. The procedure ends when a number that passes all
t
t
tests is found. Let
p
k
,
t
p_{k,t}
denote the probability that such a number is composite. The authors above have shown that
p
k
,
t
≤
4
−
t
p_{k,t}\le 4^{-t}
when
k
≥
51
k\ge 51
and
t
≥
1
t\ge 1
. In this paper we will show that this is in fact valid for all
k
≥
2
k\ge 2
and
t
≥
1
t\ge 1
.