The supervisor reported the misbehavior, but the personnel committee ignored the evidence and refused to terminate the employee.There is an exception (there's always an exception): If the two independent clauses (compound sentence) are short and closely related, you may omit the comma:
The assistants did the research and the manager wrote the report.Momentary Ambiguity
The robber tied up the mother and her son ran for help.The words her son could serve as the object of tied up. To prevent momentary ambiguity, use the comma:
The robber tied up the mother, and her son ran for help.Three or More Independent Clauses
The employees filed their complaint, the committee heard the arguments, and the company awaited the final decision.