Suppose a gambler has an initial fortune in (0,1) and wishes to reach 1. It is known that, for a subfair red-and-black casino, the optimal strategy is always to bet
min
(
f
,
1
−
f
)
\min (f,1 - f)
whenever the gambler’s current fortune is f. Furthermore, the gambler should likewise play boldly if there is a house limit z which is the reciprocal of a positive integer; i.e., he should bet
min
(
f
,
1
−
f
,
z
)
\min (f,1 - f,z)
. We show that if
1
/
(
n
+
1
)
>
z
>
1
/
n
1/(n + 1) > z > 1/n
for some integer
n
≧
3
n \geqq 3
or if z is irrational and
1
3
>
z
>
1
2
\frac {1}{3} > z > \frac {1}{2}
, then bold play is not necessarily optimal.