Does God choose who will be saved?

From Gospel Translations

In What We Believe About the Five Points of Calvinism, John Piper writes: 

If all of us are so depraved that we cannot come to God without being born again by the irresistible grace of God, then it is clear that the salvation of any of us is owing to God's election. Election refers to God's choosing whom to save. It is unconditional in that there is no condition man must meet before God chooses to save him. Man is dead in trespasses and sins. So there is no condition he can meet before God chooses to save him from his deadness. We are not saying that final salvation is unconditional. It is not. We must meet the condition of faith in Christ in order to inherit eternal life. But faith is not a condition for election. Just the reverse. Election is a condition for faith. It is because God chose us before the foundation of the world that he purchases our redemption at the cross and quickens us with irresistible grace and brings us to faith.

Listen to John Piper discuss this biblical teaching more fully--together with the great hope that it gives to the cause of evangelism.