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Thank you for giving me feedback about how my question was ambiguous. Sorry for the confusion, and I reformulated my question.

Reformulation: Let's say a black box is giving me a real number every second in a truly random way.

By truly random, I mean that there is no way for one to predict what number will come next.

But I don't yet know if the black box is giving me numbers in a random way or in some way with a pattern.

Is there any way for me to prove that the black box is giving me numbers in a truly random way?

mathlearner98
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  • Ask Eric how he generated the numbers. – mjw May 26 '21 at 18:34
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    if he is just thinking of them off the top of his head then it is likely neither, as most people are unknowingly influenced by other factors, but I will try and find a source for this. Also computers are not able to generate truly random numbers, so they use arbitrary numbers found in their environment and run them through a program to generate "pseudo random numbers" – Henry Lee May 26 '21 at 18:37
  • what if we assume that Eric is able to give numbers in both truly random way and pseudo random way? So that we also assume that Eric is giving numbers only in either way, but not both. – mathlearner98 May 26 '21 at 18:40
  • dear mathlearner98; this is a good question (+1), but I have two recommendations for your post. first, I would recommend replacing "Eric" in your post with a "black box". otherwise, as @HenryLee points out, your question becomes muddled with psychological considerations. second, it would be helpful if you could give your definitions of "random" and "pseudorandom" in the post; for instance, when you say random, what probability distribution are you thinking of? – Atticus Stonestrom May 26 '21 at 19:32
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    The only source of "true" randomness we know is the selection of a multiverse as wavefunctions decohere... i.e. wavefunction collapse or quantum randomness. Everything else is, to be pedantic, pseudorandom a.k.a. random in the Bayesian sense of lacking information about how it was generated and/or how to predict the next number in the sequence.

    Eric's numbers are random because he is affected by quantum physics. His numbers are also pseudorandom in the sense that there is an internal biological (neurological) process by which he is selecting numbers, but that process is unknown.

     – Myridium May 26 '21 at 19:34
  • Practically speaking, you could estimate the random distribution of numbers Eric is giving you by crunching statistics on a large (ordered) set of those numbers. No matter how Eric is giving you numbers, they are still random, so your question seems to be moot. You can test whether the distribution satisfies certain properties, like being uniformly distributed in a finite interval. – Myridium May 26 '21 at 19:38

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