For a given CDF compute $\Bbb P(X^2 \in A)$
Solution 1:
It should be $F(1)-F(\frac{1}{3})+F(-\frac{1}{3})-F(-1)$, not $F(1)-F(\frac{1}{3})+F(-1)-F(-\frac{1}{3})$,
though in this particular case they happen to be te same.
Finally, $\frac{1}{2}-\frac{1}{3}=\frac{1}{6}$, not $\frac{1}{5}$.