When choosing a replacement for Judas, there are two candidates, and in Acts 1:26, we read (ESV):
And they cast lots for them, and the lot fell on Matthias, and he was numbered with the eleven apostles.
Was this like a vote, or was it like a coin-toss? How would it have been carried out?
Casting lots in the ancient world was more akin to drawing names from a hat than a vote. The names of the candidates would have been written down (or something done to mark the lots), and then would be drawn from a vase, urn, or other large item. They might have been dumped out with the one landing first being named the choice.
However, research has also shown that casting lots in the ancient world could be a multi-cast affair. The lots would be cast repeatedly until the same lot fell out three times in a row. Only then was the lot seen as showing God's will. If the lots were cast ten times without a three-in-a-row, then it was deemed that God had selected neither choice.
It should be noted that the Apostles did pray that God show them his choice before they cast lots.
Proverbs 16:33 speaks of casting lots into the lap and the result being up to God (suggesting God can and will reveal His will even through what may be perceived as a random selection). This can be most clearly seen in Jonah 1:7.
As for whether it is a vote or randomness, the NET, BBE, and NLT suggest it is random whereas almost all other major translations/paraphrases just use the ambiguous "lot". Only the paraphrased version The Message suggests voting. Leviticus 16:7-8, I would say, suggests randomness as well as only Aaron was to make the decision and he would not need to vote amongst himself. Not to mention that the text makes no mention of a standard for selection to determine which of the two goats is "more suitable" for sacrifice versus being a scapegoat.
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.
