Amazingly, we have an experiment that shows that this is not the case, if you make some pretty reasonable assumptions. It's based on Bell's theorem. Your statement is very intuitive, and the fact that it may be wrong is very surprising.
Not everyone agrees on what is "reasonable". I have no problem giving up locality, but "true" randomness (i.e., information generated without an algorithm) seems like a philosophical cop out.
Note that something can also be globally deterministic but indeterministic from the perspective of a subsystem. In this sense, the universe would appear random as far as we're concerned, but not random to whoever is simulating the universe (this is often called superdeterminism, but I think that's a silly word—the universe is either deterministic or it isn't).
The probability of an event depends on your knowledge. Whether something is deterministic is a question of probability. Hence whether something is deterministic can depend on your knowledge. If the universe is deterministic to its simulator but random to us, I would call it random. Besides this difference in terminology, I think we agree.
Do you agree that the probability of an event depends on your knowledge? For example, what is the probability that the bottom card of a shuffled deck is an ace? Peek at the top card, it's an ace of spades, now what is the probability?
Would you agree that something is determined iff its probability is zero or one? I'm guessing the disagreement is here - what definition would you use?
One concerns a lack of knowledge, like in your cards example. In that case, the probability is an expression of your state of understanding of the deck, and is not a property of the deck itself. The physical details of the deck and the situation it is part of could be entirely deterministic and we could still talk about this kind of probability. In this case your knowledge is independent of whether it is probabalistic or not.
The other is whether the universe contains fundamental randomness, such that you could say it is literally "probabalistic". And in this case whether that is true or not is independent of our knowledge of the probabilities.
There's a third kind -- indexical randomness, or not knowing who you will be -- that appears in many-worlds. Many worlds is in some sense completely deterministic. Yet with indexical uncertainty you still cannot possibly ever know which way you will see a photon go in a half-silvered mirror. Is that fundamental randomness or not?
Or to give a slightly better example, suppose you're playing russian roulette with a six-shot revolver. You empty the revolver of bullets, then put one bullet in, spin the barrel, point it at your head and fire.
What is the chance that you're shot? Is it 1/6? What about if you open it up and look in the barrel to see if there's a bullet there?
You are right, according to quantum mechanics. If you have a pair of electrons in a product state and measure the spin of one of them, you instantly know what the "other particle's" spin is going to be (if you knew what the initial state was). When you "peek at a card", you "make a measurement" and alter the system.
For anything that exists outside of a light-cone around the first electron, a measurement of the second electron will be truly random from the measurer's perspective.
Not everyone agrees on what is "reasonable". I have no problem giving up locality, but "true" randomness (i.e., information generated without an algorithm) seems like a philosophical cop out.
Note that something can also be globally deterministic but indeterministic from the perspective of a subsystem. In this sense, the universe would appear random as far as we're concerned, but not random to whoever is simulating the universe (this is often called superdeterminism, but I think that's a silly word—the universe is either deterministic or it isn't).