Between the probability of union and intersection, it's not clear what you're supposed to compute. (I would guess it's the probability of union.) But we do know that
[tex]P(A\cup B)+P(A\cap B)=P(A)+P(B)[/tex]
For parts (a) and (b), you're given everything you need to determine [tex]P(B)[/tex].
For part (c), if [tex]A[/tex] and [tex]B[/tex] are mutually exclusive, then [tex]P(A\cap B)=0[/tex], so [tex]P(A\cup B)=P(A)+P(B)[/tex]. If the given probability is [tex]P(A\cup B)=0.55[/tex], then you can find [tex]P(B)=0.15[/tex]. But if this given probability is for the intersection, finding [tex]P(B)[/tex] is impossible.
For part (d), if [tex]A[/tex] and [tex]B[/tex] are independent, then [tex]P(A\cap B)=P(A)\cdot P(B)[/tex].
