Parrondo’s coin-tossing games comprise two games,
A
A
and
B
B
. The result of game
A
A
is determined by the toss of a fair coin. The result of game
B
B
is determined by the toss of a
p
0
p_0
-coin if capital is a multiple of
r
r
, and by the toss of a
p
1
p_1
-coin otherwise. In either game, the player wins one unit with heads and loses one unit with tails. Game
B
B
is fair if
(
1
−
p
0
)
(
1
−
p
1
)
r
−
1
=
p
0
p
1
r
−
1
(1-p_0)(1-p_1)^{r-1}=p_0\,p_1^{r-1}
. In a previous paper we showed that, if the parameters of game
B
B
, namely
r
r
,
p
0
p_0
, and
p
1
p_1
, are allowed to be arbitrary, subject to the fairness constraint, and if the two (fair) games
A
A
and
B
B
are played in an arbitrary periodic sequence, then the rate of profit can not only be positive (the so-called Parrondo effect), but also be arbitrarily close to 1 (i.e., 100%). Here we prove the same conclusion for a random sequence of the two games instead of a periodic one, that is, at each turn game
A
A
is played with probability
γ
\gamma
and game
B
B
is played otherwise, where
γ
∈
(
0
,
1
)
\gamma \in (0,1)
is arbitrary.