Here’s the story. Suppose I wish to pre-roll a large number of results from the roll of 3 six-sided dice (perhaps I do not wish to bring the dice with me, for example). Say I roll one result, write it down on a blank card, and then generate another result.
If I have, said, recorded down 30 results, and stop for the day. The next day I roll the dice another 10 times and hence obtained 10 more numbers. Is it valid to add those 10 news results together to the set of yesterday’s results?
A variation: say I am using a computer to generate the numbers, using a pseudo-number algorithm. I am using a very good one, say from http://www.random.org. I generate one set of 30 results, and another set of 10 results … can I combine them together and derive the probability of each number from them?
In the first case, I can’t see why you can’t. Assuming it is as truly random process as you can manage rolling the dice.
In the second case there is a subtle issue. If you use a psuedo random number generator, it will require a seed. You must be sure that you don’t use the same seed when you begin generating numbers again. Otherwise you will repeat the number sequence.
All that matters is for any true random sequence is that there is essentially no mechanism to predict the next value based upon what has gone before. Rolling a die pretty much always meets that. A pseudo random number generator violates that ideal in interesting ways, but the most important one is if you start with the same seed. The goodness of a PRNG is measured by looking at statistics that relate to distribution of results, and time to repeating. If you know the parameters to the algorithm you can always predict the next number (after, all you do is do the same calculation the generator uses). Thus, PRNGs are no use for security, unless you add an additional mechanism to inject real entropy into them. But a determistic PRNG is exactly what you need for a great many scientific purposes.
What you say about pseudo-random number generators is true, but random.org uses a variety of genuine noise sources as input, so they’re anywhere between semi-non-pseudo and not pseudo at all.
To the OP’s first question, think of it this way: How would the dice know that you’ve gone to sleep?