(A) "I borrow two books from the library?" → This asks for permission, so "May I borrow..." is appropriate → (II)
(B) "Are you sure that you run the business by yourself?" → This questions ability/capability, so "...that you can run..." is correct → (IV)
(C) "You follow the traffic rules to avoid accidents." → This expresses moral obligation/strong advice, so "You ought to follow..." is appropriate → (I)
(D) "I thought he be at home to help me with my problem." → This expresses past expectation/conditional, so "...he would be at home..." is correct → (III)
Hence, Option 2: (A) - (II), (B) - (IV), (C) - (I), (D) - (III) → "May" for permission, "can" for ability, "ought to" for obligation, and "would" for past expectation perfectly match the contextual requirements of each sentence → correct