“The lot came out for the children of Joseph.” Joshua 16 and 17 describe the inheritance of Joseph, the son of Jacob and Rachel (Gen. 30:22-24) as it is represented by his two sons Ephraim (Josh. 16) and then Manasseh (Josh. 17). Joseph is the firstborn of Jacob through Rachel, his first and beloved wife (although he was tricked out of being with her at first) which may be one reason Joseph received a double portion via Ephraim and Manasseh (cp. Deut. 21:15-17).
When Moses sent spies to search out the Promised Land (Num. 13:1-25), Joshua was the spy from the tribe of Ephraim, and he and Caleb (from Judah) were the two spies who believed that with God’s help Israel could conquer the Promised Land (Num. 14:6-9). As with Judah (Josh. 15:1), there was a “lot” involved in giving the inheritance. So the territory was Yahweh’s decision.
Joshua 1:1-4 describes the southern border of Ephraim. Joshua 1:5-10 describes the other borders, but in general the descriptions for Ephraim seem more haphazard and are less detailed than the description of the border of Judah. In describing the border of Ephraim, large territories or distances are covered in a short phrase (cp. the northern border of Benjamin; Josh.18:12-14).
“the waters of Jericho on the east.” This is almost certainly referring to the spring at Jericho now called Ein es-Sultan that was later healed by Elisha (2 Kings 2:19-22).
“going up from Jericho into the hill country of Bethel.” This is a fairly non-descript border, covering some 15 miles between points. This border is probably following a wadi, most likely the Wadi Auja.