Based on my understanding, a transcendental number is a number that is not computable, or cannot be generated by an algorithm. Let's say that (theoretically) I randomly generate a number that is infinitely long with dice. Since it is extremely unlikely that a computer algorithm would be able to generate that number, then I could conclude that I generated a transcendental number with a specific set of steps (which is the definition of an algorithm).
Obviously, there's something wrong since transcendental numbers should not be able to be generated through an algorithm. What's flawed about the logic presented above? Or is my understanding of transcendental numbers, computable numbers, or algorithms wrong?