If Jesus Foreknew Judas Would Betray Him, Why Did He Chose Him As An Apostle?

If he had never  been an Apostle he could not have fulfilled the prophecy which was based on the future reality. Jesus made no mistakes. He knew what He would do and what Judas would do.

God does not need "faith" or "hope". He is the originator and object of all true faith and hope. "Faith is the substance of things hoped for, the evidence of things not (yet) seen." Heb.11:1. 
God already owns everything in the universe, both "substance and spiritual", and hopes for nothing. He doesn't "hope", He "knows" in advance. He is our hope.    "....God our Savior, and Lord Jesus Christ, which is our hope." 1 Tim.1:1.   Instead, He has certainty of knowledge, all knowledge, omniscience. He knows that He knows that He knows everything. He does not need faith or hope, he has omniscience, omnipotence, omnipresence, the universe and every grain of sand, breath of air and drop of water in it. Irreversible, eternal perfection requires no hope or faith. It is absolute, infinite.

The Christian believer's faith is a mustard seed off of our Creator's infinite mustard tree of omniscient knowledge and omnipotent power. True faith and hope are unknown to the unbeliever, but the faith and hope of eternal life abides in every believer. 
"... Jesus Christ: whom having not seen, ye love; in whom, though now ye see him not, yet believing, ye rejoice with joy unspeakable and full of glory." 1 Peter 1:8. 
This is a mystery known only by believers, in that, "faith" converts "hope", into spiritual certainty, the "substance" of reality, before the fact, as if it were already seen and possessed... and, it is, in fact. 

Prophecy, which is simply the foreknowledge of God, of future events, given to His people through His prophets, is also certain evidence of things to come, in God's own time. True prophecy, recorded in the Bible, is certain "evidence" of future events revealed before they happen. God's prophecies, are based on God's foreknowledge of the certain fact of their eventual fulfillment, not the other way around. The fact is not based on the prophecy of it; Prophecy is based on the future fact of it. Prophecy simply reveals what is going to happen before it happens, but would happen anyway even if it had not been prophesied. God pre-knew everything He would do and what each of us would will to do.

If this be true, why did Jesus Christ choose Judas as one of His 12 Apostles, if He knew what Judas would do? Jesus knew exactly what Judas would do. Christ was with God throughout eternity and was one with God. God and Christ have known from forever that humans would be created, would sin and become fallen sinners. He knew that He would send Christ, in to the body of Jesus, to die for the sins of  all who would believe, and that Judas would betray Him. He knew all of this from eternity past and in the "Word of God", He gave prophecies of that betrayal fact even in the Old Testament, in Psm.4:9, Psm.109:8, and Zec.11:12-13. It was destined because God knew it, and let it happen as a negative force, of men's will, to contrast good from evil, God from the Devil, light from darkness, love from hate, life from death and to emphasize the divine positive of the eternal Gospel. Jesus also knew that, till the end of time, there would be traitors in and out of His churches whom Christians would have to deal with. The negative makes the positive even sweeter. Therefore, Rev.13:8 declares that Jesus was, "The Lamb slain from the foundation of the world."

If all of creation, including the human race, was only good, pure, harmless and painless, then holiness would simply be universally normal and not exceptional, but common, the only reality. But the negative gives emphatic contrast and increased emphasis to God's salvation positive.

Yes, Jesus knew Judas, that he would betray Him to be crucified.
"Jesus said unto them, the Son of man shall be betrayed into the hands of men." Matt.17:22.
"And as they did eat, he said ... one of you shall betray me."
Matt. 26:21.
"He that dippeth his hand with me in the dish, the same shall betray me." Matt. 26:23.
"Rise, let us be going: behold, he is at hand that doeth betray me." Matt. 26:46.

Judas was simply who and what he was... and chose to be.  Jesus knew it and was not surprised by it, and didn't expect him to be saved, but chose him because of who he was, not by mistake. It is also true that if Judas had sincerely repented of his betrayal action, and confessed Christ, he could have been saved, but Jesus would also have pre-known that too. But He knew Judas would not repent, only "regret" after the fact, and commit suicide. By Judas' presence, the force of the evil negative increased the emphasis on the holy positive of God's planned redemption.
"And we know that all things work together for good to them that love God, to them who are the called according to his purpose." Rom.8:28

The story would not be complete without it happening exactly as it did. He also knows exactly what you and I will, and will not do, but we are still responsible for making those decision and choices. RB