Question:How does our tradition explain the paradox of free choice?
Answer:This issue becomes difficult to discuss because we are often confused with just what the question is. As soon as we hear one question being answered, another one is nagging us.
So let us first distinguish and list those questions:
1) Determinism: Isn't everything predetermined by the mechanics of the universe?
2) Robotism: G-d knew exactly what I was going to do when He made me this way. I'm just a programmed machine. How can I be blamed for being what I am?
3) Prescience: Since G-d knows the future, what choice do we have in it?
4) Omnipotence: G-d wants something to happen and it happens. So how could I possibly choose to do something He doesn't like? Who's more powerful, after all?
5) Oneness: Since there is nothing else but His Oneness, what room is left for us to make any difference?
6) Primal Cause: If G-d is the Primal Cause, doesn't the buck stop there?
Each question has its particular answer. So let's deal with these one at a time (since each question picks up were the previous one leaves off, it's best to read them in order):
Isn’t everything predetermined by the mechanics of the universe?
The Short Answer:
G-d generally delivers in rather predictable patterns. Spring follows winter. Horses give birth to horses. If an object moves unimpeded at 60 centimeters per minute, it will transverse 120 centimeters in two minutes. A baby born of two blonde parents will likely be blonde. But, since the whole thing stems out of His unbridled imagination, within those patterns, G-d has unlimited flexibility. And, of course, He always reserves the option to do the totally unexpected.
A Little Longer Answer:
Determinism was an idea that evolved over many ages, reaching its pinnacle in the 19th century. The universe was seen as a big machine functioning along the same principle of cause and effect as a Swiss clock. It was married to another idea called Reductionism, which states that if we know what the basic particles of the universe are doing, we will be able to explain the behavior of all the big things that are made from those particles. That would include chemicals, plants, animals, and even us human beings.
Determinable systems had been observed and measured for millennia -- and more precisely since the time of Galileo. The assumption that all systems, including the most complex, are determinate was never demonstrated. Neither was reductionism. Neither was the assertion upon which it rested, that the universe is built of irreducibly small particles with predictable behaviors. These were all no more than assumptions made on the basis that, well, they sounded nice. Machines made by humans work this way, so why shouldn't the universe? Call it "creating G-d in our image".
When scientists developed tools precise enough and methods rigorous enough to examine the universe close up, they discovered to their amazement that not one of these three assumptions had any basis to it. On the contrary, the uncertainty principle has been universally accepted by physicists. A popular interpretation of this principle -- with increasing evidence -- is that human consciousness plays a major role in determining reality. There are even respectable scientists who state categorically that things happen because we observe them to happen. (See Eugene Wigner, Symmetries and Reflections, chapter 13; See also Dr. Naftali Berg's article in Bor HaTorah Vol. 9, pp. 34-49.)
Those who work in the field of chaos theory and complex systems point out that a small change in a single atom can ripple through the cosmos, effecting major impact on the macrocosm. Since electrons are currently understood to be indeterminate, the state of the cosmos must be as well. We can provide general predictions, but the details will always escape us. Not because they are too difficult to determine, but because they simply are not determined by their precedent.
Of course, there is far from a consensus on the nuances of these matters. But throughout the panorama of positions, all will agree on the following: If anyone wishes to believe that the universe steps along a single, predetermined path, he must do so as a matter of faith, not of science.
Why don't I just let the mechanics speak for themselves? These words were spoken before a representative gathering of scientists in 1986:
"I have to speak on behalf of the broad global fraternity of practitioners of mechanics. We collectively wish to apologize for have misled the general educated public by spreading ideas about the determinism of systems satisfying Newton's laws of motion that, after 1960, were to be proven incorrect." 
The Torah view is clear and unequivocal: It is not the laws and state of the universe that determine what will be next, but a Supreme Consciousness that transcends all this scheme. It is just that this Consciousness generally prefers to get His way within the boundaries of consistent patterns. That consistency is what appears to us as the Laws of Nature.
Furthermore, this Supreme Consciousness is not to be viewed as some outside force who happened to come across a universe and decided to muck about with it. Rather, that Consciousness is the perpetual origin of all that is. At every moment, all matter and energy is regenerated into being out of the void, something like a movie being projected onto the screen -- except that in this case, time and space are also perpetually renewed. In the words ofMaimonides, "If He would cease the state of being, all else would not be." And as Rabbi Schneur Zalman further explains, "If the forces G-d uses to bring the cosmos into being would cease for a moment and return to their source, the entire cosmos would be absolute nothingness, just as though they had never been, returning to the void that preceded the six days of creation."
Today, this view as well has been adopted by many respected scientists. In the words of an outstanding astrophysicist, Professor William Stoeger, "The act of bringing something out of nothing (as long as the something is contingent) is needed at every temporal and spatial point. The primary cause cannot cease to act in support of secondary, contingent causes, processes, or entities -- cannot cease to be influential and effective -- without the universe, which is contingent, slipping back into absolute nothingness." (Cosmos, Bios, Theos, pg. 259, ed. Margenau and Varghese, Open Court, Chicago, 1992)
The practical difference between this scheme and the mechanical view of the cosmos is twofold: First of all, there is nothing stopping this Original Consciousness from bending or even breaking the rules of consistency on occasion. Secondly, the so-called Laws of Nature (i.e. rules of consistency) are not so rigidly linked -- any situation can branch into any one of a wide range of consequences. Its up to the Big Consciousness to choose at every point. Fundamentally, it comes down to this: The world is not an artifact of blind, dumb laws. It is a continuously deliberate act of will and intelligence.
How does He choose? According to how we choose. This is what the Torah repeats over and over: "G-d says, If you do like this, then I will do like that." AsKing David described the relationship, "G-d is your shadow."
Not everything is in the hands of our free choice. Winter will follow fall, the sun will rise in the east, some will be born smart and others with more limited talents. But the things that matter in life -- meaning, what you do with what you've been given, and what will come as a result of that -- that's left up to you.
On to the next question:
G-d knew just what I was going to do when He made me this way. I'm just a programmed machine. How can I be blamed for being what I am?