What does it means for a sample to be random?

Question

For a sample to be random, I thought it means that every member has an equal chance of being selected.

However, it seems like it is insufficient to just say the above. May I know what is the proper definition for a random sample? What am I missing?

Thanks.

Answer 1

If you're sampling $m$ items (without replacement) from a finite population of items, you want every set of $m$ distinct items in the population to have equal probability of being your sample. If sampling with replacement, you want every ordered $m$-tuple to have equal probability of being your sample.

Answer 2

Technically, "random" does not simply mean that each element is "equally probable" or "equally likely" to appear. That is just one particular distribution: the uniform distribution. Instead, samples can be "random" even if there probabilities are unequal. For example, a "weighted coin" which has a 60% of being HEADS, 40% chance of being TAILS is technically random.