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If I take some sequences of random numbers generated by a random number generator with uniform distribution, will the resulting sequences be uniformly distributed as well?

By example, if I have a generator that returns 1, 2, or 3, what are the probabilities to get [1, 1, 1] and [1, 2, 3]?

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It depends on whether the generator outputs independent variates or not. (I assume from your question that it outputs identically distributed variates.) If it is independent and identically distributed than the answer is positive: every sequence will have a probability of $\frac{1}{k}\cdot\frac{1}{k}\cdot\ldots\cdot\frac{1}{k}=\left(\frac{1}{k}\right)^p$ (where $k$ is the number of outcomes of a single experiment, all being equally likely, $p$ is the length of the sequence, i.e. the number of experiments). In your example $\left(1/3\right)^3$.

Hence, the sequences are also uniformly distributed. Note that I could have used product due to the independence.

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