A philosophical question regarding random numbers....

[R. E. Nettleton]

A number can be random even if limitations are applied to the outcome - e.g. selecting a random integer between 1 and 5 restricts the outcome to one of 5 numbers, but the outcome is still random. The same would be true of between 1 and 2; although there are heavy restrictions, an unbiased machine will output one number randomly.

If we simply go one limitation further, and restrict the randomly generated number to being, for example, between 1 and 1 (i.e. 1), is the number generated still random? Of course, the output can only be one number, so in that sense it is determined - but at the same time it is still determined randomly, just with hyper-restrictive limitations, for the generating machine remains is still selecting without bias.

 

[R. E. Nettleton]

*A correction to the first sentence: the outcome would be restricted to 1 of 3*

 

[Mentallic]

True randomness means that it cannot be predicted whatsoever. Since the number can be predicted with absolute certainty, it isn't random.

 

[Mark44]

A number can be random even if limitations are applied to the outcome - e.g. selecting a random integer between 1 and 5 restricts the outcome to one of 5 numbers, but the outcome is still random. The same would be true of between 1 and 2; although there are heavy restrictions, an unbiased machine will output one number randomly.

There aren't any integers between 1 and 2. If what you meant was that 1 or 2 would be randomly chosen, then the outcome would be random.

 

[fresh_42]

One out of one can still be regarded as random. The underlying distribution in this case is a trivial one: 100% on 1, 0% elsewhere, which is discreet. Rather boring, but not in contradiction to the definition of a random variable.

 

[HallsofIvy]

There is no such thing as "a random number". That is a badly chosen way of talking about "randomly chosen numbers".

 

[Hornbein]

A number can be random even if limitations are applied to the outcome - e.g. selecting a random integer between 1 and 5 restricts the outcome to one of 5 numbers, but the outcome is still random. The same would be true of between 1 and 2; although there are heavy restrictions, an unbiased machine will output one number randomly.

If we simply go one limitation further, and restrict the randomly generated number to being, for example, between 1 and 1 (i.e. 1), is the number generated still random? Of course, the output can only be one number, so in that sense it is determined - but at the same time it is still determined randomly, just with hyper-restrictive limitations, for the generating machine remains is still selecting without bias.

Sure, but this is what is called a "degenerate case." Something uninteresting, but allowed because it is too much trouble to exclude it.

"Random" just means "unpredictable." Though the word is often used for "choosing with equal probability for each case."

 

[Josh S Thompson]

If it is randomly generated than it is random. It could be one but it could also be one. I think its degenerate case.

 

[HallsofIvy]

If it is randomly generated than it is random. It could be one but it could also be one. I think its degenerate case.

Sorry, but I can't make heads or tails out of this: What do you mean by "It could be one but it could also be one"? And what do you mean by "degenerate case"?