Recover A from a disjoint region before using independence

$P(A)=P(A\cup B)-P(A'\cap B)=3/4-2/15=37/60$. This first step is a set decomposition and does not require independence.

Independence gives $P(A\cap B)=P(A)P(B)$. Thus $3/4=37/60+P(B)-P(A)P(B)$, or $2/15=P(B)[1-P(A)]=P(B)(1-37/60)$. Therefore $P(B)=(2/15)/(23/60)=8/23$.

$P(A'\cap B)=P(B)-P(A\cap B)=P(B)-P(A)P(B)$. Hence $2/15=P(B)[1-P(A)]=P(B)(1-37/60)$, again giving $P(B)=8/23$.