In statistical terms a random sample is a set of items that have been drawn from a population in such a way that each time an item was selected, every item in the population had an equal opportunity to appear in the sample. In practical terms, it is not so easy to draw a random sample. First, the only factor operating when a given item is selected, must be chance.
If, for example, numbered pieces of cardboard are drawn from a hat, it is important that they be thoroughly mixed, that they be identical in every respect except for the number printed on them and that the person selecting them be well blindfolded.
Second, in order to meet the equal opportunity requirement, it is important that the sampling be done with replacement. That is, each time an item is selected, the relevant measure is taken and recorded. Then the item must be replaced in the population and be thoroughly mixed with the other items before the next item is drawn. If the items are not replaced in the population, each time an item is withdrawn, the probability of being selected, for each of the remaining items, will have been increased.
For example, with the illustrated population, the initial probability that a given item will be selected is 1/9. If, however, an item is drawn and not returned before drawing a second item, the probability that a given item will be drawn will have been increased to 1/8. Of course, this kind of change in probability becomes trivial if our population is very large, but it is important to recognize the principle illustrated here, to fully understand the concept of a random sample.
It is also important to recognize that when sampling with replacement, it is possible for the same item to appear more than once in a sample and it is possible to draw a random sample that is larger than the population from which it came. Notice also, that it is possible to draw as many random samples as we like from a give population. The key idea here is that we either sample with replacement or we draw our samples from a population that is so large that the withdrawal of successive items changes probability by an amount that is too small to be of concern.