Is it possible to check if a sequence of numbers (1 to 10) have been generated randomly or by a human? [closed]

Question

Let's suppose to have sequence like "10 1 2 4 5 3 2 1 2 3 10 1" and so on... It is possible to understand if it has been generated randomly (by a program) or by a human? Do they correspond to a kind of distribution?

Answer 1

Despite the comments, your question sits at the entryway to what has become an exciting new field of research.

In classical probability theory, if one were to flip a coin a certain number of times, then any two particular patterns of flips would be equally likely, and so in this sense, it doesn't make sense to say that one sequence is more random than another, or that any particular sequence is "random."

Yet, we seem to have an intuitive sense that some sequences look more random than others.

The subject known as algorithm randomness (see also Algorithmic Randomness and Complexity, Downey, Rodney G., Hirschfeldt, Denis R.) aims to give legs to that intuition, by identifying various criteria of randomness that a particular (infinite) sequence may or may not exhibit.

For example, an infinite binary sequence is said to be generic, if it is a member of every computable co-meager set. A sequence is random, if it is an element of every suitably computable measure-one set. And so on for many notions of randomness that have been proposed. The subject is concerned with sorting out the relations between these various randomness notions.

Answer 2

Recapping some aspects of @Joel David Hamkins' answer: "probability" by itself is not enough to explain "randomness", as it turns out. After all, all possible sequences of 100 heads-or-tails are equally likely, but some would be more suspect of being due to an unfair coin or other human intervention. To my perception, the issue is not about probability, but about "complexity" (e.g., Kolmogorov-Solomonov-Chaitin notions...).

An approximation to what "complexity" is about can be modeled by "compressibility", namely, if a thing admits a description much smaller than itself, it is not "random". Of course, the notion of "description" is context dependent, etc. But at least the intuitive notion of this seems (to me) to exactly address the true notion of "randomness".

So, if a file is not compressible by the standard compression algorithms, it is likely that it is either already compressed or is well-encrypted, for example. Indeed, one might appraise the quality of an encryption scheme by trying to compress the ciphertext: if it does not compress well, it is (at a low level) perceived as "random", which is desirable.

Answer 3

If you know the distribution of the random number generator, you can compute the p-value of your sequence for the Kolmogorov-Smirnov test on that distribution (e.g., ks.test on R). The closer the p-value is to 1, the more likely the data is from a generator. If you don't know the distribution but do know that the generator is independent across elements, you can compute the p-value for that assumption (e.g., Pinkse, 1998; Kulperger and Lockhart, 1998; Ghoudi and Kulperger, 2001). Basically, if you know any other characteristics of the generator, you can compute the p-value for those characteristics.

Alternatively, if you have tons of example sequences created by humans and generators, you can make a neural network learn to distinguish them (e.g., with Tensorflow on Python).