Right off the bat, I see a number of glaring errors. Two seem to come from not paying close enough attention to the drawing, and the other from a fundamental misunderstanding of terms. First, there are **four** green squares out of eleven total shapes, so \(\displaystyle P(\text{green AND square}) = \dfrac{4}{11} \ne \dfrac{2}{11}\). Second, there are five green shapes and two yellow squares, making **seven** shapes that are either green or square (or both), thus \(\displaystyle P(\text{green OR square}) = \dfrac{7}{11} \ne \dfrac{6}{11}\). Correcting these two errors, we see that:

\(\displaystyle P(\text{green}) + P(\text{square}) - P(\text{green AND square}) = \dfrac{5 + 6 - 4}{11} = \dfrac{7}{11} = P(\text{green OR square})\)

so the supposition holds.

Finally, you appear to have some confusion about what's going on. In this particular instance, whether the two events are independent doesn't matter at all, because you're only picking one shape. Rather, what you need to ask yourself is: Are the events mutually exclusive? That is to say, can both happen at the same time? Obviously the answer to that question is yes - there are two green squares in the drawing.

Strictly speaking, the formula P(A OR B) = P(A) + P(B) - P(A AND B) always holds, but when the two events are mutually exclusive (i.e. both *cannot* happen at the same time), this necessitates that P(A AND B) = 0, so the formula simplifies down to P(A OR B) = P(A) + P(B).