In the New Testament, in the Bible, did Jesus make a mistake choosing the twelve apostles?
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I do not believe this question can be solved objectively—with facts; scholars and philosophers would answer based on Biblical writings as the Bible is theological and faith-based in nature, rather than scientifically-based, so an answer would be subjective.
In the New Testament of the Bible, Mark, Chapter Three, provides a description of Jesus' selection process in selection his twelve disciples. The careers of some of these men are unknown, but we are told that one was a tax collector, and four others were fishermen.
Peter, Andrew, James, and John were two sets of brothers. They were helping run a family business when Jesus invited them to join him. Matthew, who was a tax collector, would have been extremely disliked among his community, and to an objective mind might seem an unusual choice, but none of these men were chosen because they were perfect.
The sense in all three of the gospels that include this information (Matthew, Mark and Luke) is that Jesus had a clear purpose in selecting the individuals who he would work with, and historically, twelve was not an unusual number in the Biblical sense. According to Dr. Marcellino D'Ambrosio:
[Regarding the] foundation of the people of Israel, [t]here were twelve tribes, traced back to twelve patriarchs, all sons of Jacob. Jesus had come to found a new Israel, called the Church, and so chose twelve as the patriarchs of this new family.
[In terms of] the twelve sons of Jacob...they were called to assume a particular responsibility. The same is true with the New Testament twelve. They are to be spokesmen, etc.
[Additionally,] if the twelve were called to be spokesmen for the entire Church, Peter was called to be spokesman for the twelve.
As far as Judas Iscariot—the traitor—is concerned, his presence in fulfilling Old Testament prophecies was absolutely necessary, in that his betrayal of Christ led to his crucifixion. Biblically speaking, this was part of a plan foretold many years before it actually occurred.
Biblical scholars would see no "mistake" in Jesus' choice of his twelve apostles.