Thus, choosing to believe is something we do of Free Will; knocking absolute predesitination into the ashpile.
*2 Peter 3:9
You would be sincerely wrong, since, if "all" are 'chosen' to receive salvation, then all would be saved, since "all that the Father giveth me shall come to me," and, "whoever comes to me I shall not cast out." You also didn't reply to the the other verse wherein Christ explains the reason for the unbelief of the Jews, which declares quite plainly that it was not given to them to believe at all, and that is why He told them "no man can come to me unless it is given to Him by my Father."
One final point, if you had free will, so it follows then that you can freely choose at any time, under any circumstance, one way or the other, as your will is in not any way under the bondage of your own sinful desires. But this contradicts the scripture (among many others) which declares "no man can say that Jesus is the Lord, but by the Holy Ghost" 1 Co 12:3. If your will must depend on the Holy Ghost revealing Himself to your first, then, your will is in bondage, and, at most, you can say that you only have the "freedom" to choose until the power is given to you from above. In which the question then is, "Can I refuse then?" And that answer is, again, no, for whomever Christ frees, they are free indeed, and what God hath begun in you, He sees through to completion (Php 1:6, John 8:36).
It is indeed man's duty and God's great desire that all men every repent and be saved, in the same way it is His desire that all men repent and sin no more; however, the problem is that in man's "free will," he has no desire to fulfill his duties, and only freely chooses to sin. And that is why God is the active agent in salvation, as otherwise none would be saved, as all of us are "gone astray" and "do not seek" or "understand" God (Romans 3).
But as for why God does not save one person or another, or why He does not save all, the answer is "I will have mercy on whom I will have mercy, and compassion on whom I will have compassion" (Romans 9). When God saves anyone, it is in mercy that He does so, and when He judges anyone, it is in perfect justice that He destroys them.