Given two sample spaces X and Y, the Cartesian product X × Y is the sample space { (xi, yj) | xi ∈ X ∧ yj ∈ Y }. Sample space products may be extended to any number of constituent sample spaces.
Outcomes are independent if prior outcomes have no influence over future outcomes. Independence is a tricky concept, and is unlikely to be realistic.
The Cartesian product of symmetric, independent sample spaces? is a symmetric sample space. The probability of an outcome from a Cartesian-product sample space is the product of the probabilities of the constituent outcomes, or you can assume symmetry and assign the resulting probability.
Understand what things are, but then understand why they are that way.