It is known that an optimal strategy for a gambler, who wishes to maximize the probability of winning an amount
a
−
x
a - x
in a subfair red-and-black casino if his initial capital is x, is the bold strategy in which the gambler wagers at each opportunity the minimum of his entire current capital
x
′
x’
and the amount
a
−
x
′
a - x’
required to reach the goal a if he wins the bet. If the casino imposes an upper limit L on wagers, we shall prove that the modified bold strategy of wagering
min
(
x
′
,
a
−
x
′
,
L
)
\min (x’,a - x’,L)
is optimal, at least in the important special case in which the goal a is an integral multiple of the house limit L.