Question

Suppose the dial on the spinner is spun 2 times in a row.

Answer 1

if this is your spinner the answer is

the answer is

X    P

0   1/9

1    4/9

2   4/9

Answer 2

If you spin the dial two times there three things that can happen:
1) Dial does not land in the region A
2) Dial lands once in the region A
3) Dial lands twice in the region A
The combined probability of these three events has to be 1. 
Let's calculate probabilities of each event:
$1) P=P(1,1)=\frac{2}{3}\cdot\frac{2}{3}=\frac{4}{9}\\ 2)P=P(1,2)+P(2,1)=2\cdot \frac{2}{3}\cdot \frac{1}{3}=\frac{4}{9}\\ 3)P=P(3,3)=\frac{1}{3}\cdot\frac{1}{3}=\frac{1}{9}$
If we summ all those probabilities we get 1. 
The answer would table C (lower left corner).