Final answer:
The primary purpose of King Darius' initiatives such as establishing satrapies, a national road, and a courier system was administrative efficiency in governing the Persian Empire.
Step-by-step explanation:
The primary purpose of King Darius' initiatives such as establishing satrapies, a national road, and a courier system was administrative efficiency. These reforms were implemented to address the logistical challenges of governing a large empire and ensure effective communication and control. The satrapies helped in the efficient administration of the empire by dividing it into provinces, each ruled by a satrap and a military general. The national road and courier system facilitated quick and reliable communication between the satraps and the king, enabling the centralized administration of the Persian Empire.