The text in question did not say that James or Peter, or anyone else, picked out a "lot" (singular) from a jar or sack , etc. Rather the scripture stated that they gave forth their lots. All the disciple (they, pl.) gave forth their lots (plural). The next "lot" in the verse is singular but in a collective sense.
Interestingly, the Athenians would each pick up a white or black stone, and place them in a jar. Then they would be counted by a administrator, and the "lot" (winner) would be declared. However, the scriptures are silent about any Athenian influence in the procedure. But what is known is they prayed first!
And everyone seemed satisfied with the procedure.