The problem with these two doctrines in reformed theology is that we see a certain liberty and predestination exposed in the Bible. So saying that predestination is the result of foreknowledge would not be accepted by Calvin for example.
Calvin seemed to foresee that there would be people that would argue that God "distinguishes among men according as he foresees what the merits of each will be" (Inst. III, 22, 1). Calvin, accordingly, writes against this notion, "by thus covering election with a veil of foreknowledge, they not only obscure it but feign that it has its origin elsewhere" (Inst. III, 22, 1). Calvin contests that this view of foreknowledge makes man God's co-worker in salvation, and implies that election is ratified only by man's consent. This is to make the gravest of errors because it suggests that man's will is superior to God's plan, or at the very least, implies God's plan is partially dependent on man(Inst. III, 24, 3). In refutation of this view, Calvin asserts that "this plan was founded upon his freely given mercy, without regard to human worth" (Inst. III, 21, 7 emphasis added).
This being said, predestination is not foreknowledge on the part of God, but an act of his sovereign will over his creation.
But the Bible still presents a great degree of liberty. For example, the brothers of Joseph went to him after Jacob's death to seek protection. Gen 50.20
As for you, you meant evil against me, but God meant it for good, to bring it about that many people should be kept alive, as they are today.(ESV) In this case the brothers used their liberty to plan evil, but it was the will of God that was executed. So to take the illustration of the OP in this scenario: the brothers used their freedom, but when they looked behind them they saw the sovereign God at work or predestination to save all the sons of Jacob.
The analogy I heard the most about these two doctrinal statements is: they are like the two rails of the railroad. The never join, but you still need them both to ride the train. This illustration demonstrates how some reformed theologians hold these views. They are both presented in the Bible and in the mind of God they do not conflict one another. It is not a clear-cut answer or an airtight argument, but to a certain degree it is one that does justice to the whole teaching of the Bible and does not place our view of reality over God's view of reality.
In the illustration presented by the OP: the individual makes a choice of his own liberty, but when he looks back on it he sees the hand of God guiding him in his choice. It is as if the freedom of the will and predestination was two sides of the same coin in the mind of God.
EDIT NOTE: Thanks to the question and comment of @Hammer and @Nathan Bunney which helped me to clarify my thoughts and correct an error that I left in my first response.