According to Acts 1:20-26, the disciples fulfilled the Psalm prophesy about needing to find another person to take Judas' place. It then says the disciples prayed to God about the decision after selecting two men who had been with them since the beginning of Jesus' ministry on Earth. They cast lots to make the decision, which was the equivalent of rolling dice or drawing straws, and Matthias was the one chosen. To a believer in the Bible, Matthias was chosen to be the 12th disciple because that is what God's will was. This was in accordance with a fulfillment of the Scriptures, which Peter quoted in v. 20 from Psalms 69:25 & 109:8 for replacing Judas' position. As for Biblical appropriateness of casting a lot, consider Proverbs 16:33: "The lot is cast into the lap; but the whole disposing thereof is of the LORD."
The Bible does indicate why Matthias was chosen over Barnabas - only Matthias & Joseph Justus (Barsabas) seem to be qualified for the position. Barnabas, a native of Cyprus, while being a godly Early Church believer, did not meet the criteria that Peter explained in Acts 1:21, 22. Matthias was shown to be God's choice, there being no Biblical reason to question it.
Incidentally, both Barnabas and Titus are called apostoloi (apostles) in the Greek text, which means a sent one, but they are not numbered with the original 12. Even Paul called himself "as one born out of due time. For I am least of the apostles, that I am not meet to be called an apostle, because I persecuted the church of God (before he became a Christian)." 1 Corinthians 15:8,9.
The 13th disciple or rather the replacement for Judas Iscariot was decided between two men, Joseph called Barsabbas and Matthias. The original 11 prayed for God's blessing and they cast lots to determine the chosen man. Apparently both men were held in high esteem and well qualified for this task so the outcome of casting lots (flipping a coin) could be understood to be decided by God. The chosen man was Matthias. Acts 1: 21 - 26. A Muslim man called Ahmed Al-Hasan wrote a book in which he claims to be the thirteenth disciple.
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