Exploring the Concept of Randomness in Mathematics

[Loren Booda]

Please give your definition of mathematical randomness. Consider what randomness is, as opposed to what it is not.

Does randomness or order approach probability one as the number of samples approaches infinity?

Is our universe globally and/or locally random?

Are there different degrees of randomness?

 

[dav2008]

This thread is a perfect example.

 

[Loren Booda]

I recommend your squirrel stop hitting the pipe and get back to running the wheel.

A negative answer is better than a nonsensical one.

 

[jhae2.718]

Oh, this is something I've had long arguments about.

A basic definition would be that a number is random if every single possibility has an equal probability and if there is no way to predict future values from past values.

 

[ideasrule]

My definition of random (or rather, a random distribution): if you can predict the results of a random process with better accuracy than the random distribution, then the process is not random in the way you expected. If the random distribution predicts the result of an infinite number of trials more accurately than any other model, the process is truly random, and your distribution is correct.

 

[dav2008]

My definition of random (or rather, a random distribution): if you can predict the results of a random process with better accuracy than the random distribution, then the process is not random in the way you expected. If the random distribution predicts the result of an infinite number of trials more accurately than any other model, the process is truly random, and your distribution is correct.

I think that reasoning has a slight flaw in that you use "random process" in the definition. Also, it only helps predict what is not random.

I could sit here with a coin and place it heads or tails on the table and record my decision. In the long run I'll probably have the same number of heads and tails and have a uniform distribution, but that doesn't mean it was randomly generated

 

[jobyts]

Oh, this is something I've had long arguments about.

A basic definition would be that a number is random if every single possibility has an equal probability and if there is no way to predict future values from past values.

That definition seems to be under the assumption that the time is an always forward moving variable.

 

[JaredJames]

That definition seems to be under the assumption that the time is an always forward moving variable.

Well until we have a time machine, it works for me.

 

[Proton Soup]

randomness is simply events that you lack the ability to predict.

of course there are different kinds of randomness. and plenty of tools to analyze it. not only may probability distributions have different shapes, but successive samples may have different correlations.

but at some point, it comes down to a philosophical viewpoint, such as einstein saying god does not play dice with the universe. yet, even if everything in the universe is predetermined and proceeds along some algorithmic path, it still is very random to you and me because we are incapable of predicting it.

 

[Jimmy Snyder]

42 is a random number.

 

[Pythagorean]

A random number is a number chosen as if by chance from some specified distribution such that selection of a large set of these numbers reproduces the underlying distribution.

http://mathworld.wolfram.com/RandomNumber.html

 

[jhae2.718]

http://xkcd.com/221/"

 

[jobyts]

Well until we have a time machine, it works for me.

Then it's a pseudo random number, not a pure random number.

 

[micromass]

This is an EXTREMELY interesting question. I believe that randomness does not exist in reality and that everything is determined in some way. Of course, there is no way to be certain of this statement.

Randomness does exist, however, in the mathematical world. In mathematics, we can define a random process by a collection of functions, called "random variables". The idea is that we catch this randomness somehow and make it less random by assigning probabilities to various events.
But, why should we care about randomness in mathematics if randomness does not even exist in the real world? Well, the real world is perhaps not random, but it certainly looks like it is! Thus, randomness in mathematics can offer a good approximation to the real life situation.