Confusion about series of pure Random numbers

Question

Here is a big question about random numbers: can a series of random numbers contain repetition of numbers?

I am confused about what are called pure random numbers? I suppose if we wanted to generate a series of 20 random numbers in the range 0-9 then obviously there must be repetition, but because of that, we cannot say that the series is pure random numbers, right?

Answer 1

Of course a series of random numbers may contain repetition of numbers. When you throw a purely random dice, it can also fall on the same number twice in a row!

Answer 2

Generally, we do not say that one specific sequence is random. Instead, we talk about random distributions (many possible sequences and their probabilities).

When you flip a coin, it may land heads up or tails up. Heads is not random, and tails is not random. It is the act of flipping a coin that is random.

Random sequences may be drawn (or sampled) from many different distributions. When we choose a distribution, we are often trying to model some physical process. For example, if we want to model several die rolls in order, then we may draw several integers from 1 to 6. Since the die rolls we are modeling could have repetitions, then our draws may have repetitions. The sequence [1, 1, 1] has equal probability of occurring as [2, 5, 3]. Distributions in which each possibility has equal probability are called uniform.

If we want to model shuffling cards, then there cannot be repetitions. When a physical deck of cards is shuffled, each card has exactly one location; it cannot be repeated. In this case, the sequence [1, 1, 1] cannot occur. A distribution for shuffling cards would also be uniform (every possibility has equal probability), but it would not contain sequences with repetitions.

The methods of computation used to select a sequence from the random distribution vary depending on the distribution. To create a sequence without repetition from a generator that does repeat, we can simply collect a sample from the generator, test whether it is a repeat, and, if it is a repeat, discard it and try again. This works, but there are more efficient ways to compute such sequences.

There are also distributions with unequal probabilities, which occur in situations such as finding the average of many samples of a distribution or in samples of arrival times of customers.