Step 1: Understanding the Question.
We need to pick the bank that is privately owned rather than government owned, among the four listed.
Step 2: Key Fact or Approach.
Ownership decides the category. If the Government of India holds majority shares, it is a public sector bank. If private shareholders control it, it is a private sector bank. India nationalised 14 major banks in 1969 and 6 more in 1980.
Step 3: Detailed Explanation.
Bank of India, Bank of Baroda, and Punjab National Bank were each part of the 1969 batch of nationalised banks and have stayed government owned ever since. South Indian Bank, on the other hand, was set up in 1929 in Kerala and was never included in any nationalisation round. It has continued to operate as a private sector bank with shares traded and held largely outside government control.
Step 4: Final Answer.
South Indian Bank is not a public sector bank, so option (a) is the answer.