All that says is that 1,2,1,2,1,2... is a possible uniform random sequence. It is not the only such sequence, and it is a very unlikely sequence. The notion o

All that says is that 1,2,1,2,1,2... is a possible uniform random sequence. It is not the only such sequence, and it is a very unlikely sequence. The notion of "distinguishible" here refers to an experiment where an algorithm can receive one of two possible inputs, with equal probability: a sequence of uniformly sampled numbers, or a sequence generated by a PRNG. If the PRNG is secure, the algorithm should output "1" when it receives the uniform sequence with (very nearly i.e. different by a negligible amount) equal probability as when it receives the PRNG's sequence. So, for example, if you test if even indexed elements are 1 and odd indexed elements are 2, and output 1 if both tests pass, then you will almost never output 1 for a uniform sequence; on the other hand, if the PRNG always and very frequently outputs 1,2,1,2,1,2... then you will always or very frequently output 1 when you get the PRNG's sequence as input, and thus your probability of outputting a 1 is greater in that case.