Answer
$X^2\lt X_α^2$: null hypothesis is not rejected.
There is not enough evidence to conclude that the roulette wheel is out of balance.
Work Step by Step
$H_0:$ the roulette wheel is working fine.
In this case: $P(black)=P(red)=\frac{18}{38}=\frac{9}{19}$ and $P(green)=\frac{2}{38}=\frac{1}{19}$
$H_1:$ the roulette wheel is out of balance.
Total: the pit boss spins the wheel 500 times.
Expected count of black: $500\times\frac{9}{19}=236.84$
Expected count of Red: $500\times\frac{9}{19}=236.84$
Expected count of Green: $500\times\frac{1}{19}=26.32$
$X^2=Σ\frac{(O_i-E_i)^2}{E_1}=\frac{(233-236.84)^2}{236.84}+\frac{(237-236.84)^2}{236.84}+\frac{(30-26.32)^2}{26.32}=0.58$
$k=3$. So, $d.f.=3-1=2$
$X_α^2=X_{0.05}^2=5.991$
(According to Table VII, for d.f. = 2 and area to the right of critical value = 0.05)
Since $X^2\lt X_α^2$, we do not reject the null hypothesis.
