A random sequence of numbers is one without order or pattern
Ahh… but how can you demonstrate that there’s no order? And, more to the point, if you’ve only seen r1…ri, then how do you know that there’s not a pattern once ri+1 has been seen?
In the absence of an agreement on the definition of randomness and in the absence of an agreement on an example of how to produce a random sequence, it is difficult, if not impossible, to discuss these concepts with any degree of clarity.
Then again, if we have an example of something putatively “without order or pattern”, then we have an example of something that’s likewise putatively not deterministic, don’t we? And, if so, then we have a certain sense of clarity that determinism doesn’t hold up to scrutiny… don’t we?
My observation was that most books will say that the numbers obtained from a die toss are random
I think you should examine more carefully what they mean by “random”, then.
Further there is no pattern and there is no correlation between the present numbers and the numbers obtained in the future by the die toss
Hang on, though – there’s a difference between a “series of events” and a “single event”, isn’t there? You seem to be invoking the notion of independent events rather than the notion of a deterministic event.
This confusion proceeds because the ‘proof’, as it were, of the “randomness” of a particular value that’s output by the generator seems to proceed from the possibility of predicting the i+1
th element of a series of values. That’s not what’s in play, per se, here. The question is merely whether a single value is random. You seem to be conflating the concepts.
For my purposes, the following three examples are examples of randomness:
- Tossing a fair die and listing the numbers obtained.
- Flipping a fair coin.
- White noise consisting of serially uncorrelated random quantities.
#1 and #2 could be predicted, if all the variables were understood and calculated. #3 begs the question, since it presumes “random” (name removed by moderator)uts.
None of these helps your argument.
I do not believe this to be true
Your belief doesn’t come into play. Neither does our current ability to predict events at gambling tables.
To continue the discussion there has to be agreement on what is meant by a random sequence
You can recognize that you’ve moved the goalposts, right? We weren’t talking about “sequences”…