Let n1 and n2 be the number of red and black balls, respectively, in box I. Let n3 and n4 be the number of red and black balls, respectively, in box II.
Let E3 be the event that transferred ball is red and E4 be the event that transferred ball is black.
R be the event of selecting a Red ball.
P(R)=P(E3).P(R/E3)+P(E4).P(R/E4)
⇒P(R)=13=n1n1+n2.(n1−1n1+n2−1)+n2n1+n2.(n1n1+n2−1)=n1n1+n2
only options C and D satisfies the above relation.