# Expected value without complete sample space

The book way: Suppose, we have a bag with 8 balls numbered 1-8, we want to estimate the population parameter mean. we note down the entire sample space. (1,1)(1,2).. (8,8) calculate mean of each possible pair, get the frequency plot and then find the probability of each mean. The expected value i.e sum( p(xbar)*xbar) where xbar is mean obtained from the pair and probability is from the frequency. This expected value is said to be population mean.

But are we not supposed to draw a sample and then estimate the population mean from that sample? And is a single sample enough or do we need to draw multiple samples?

My question stems from the fact that the book used 1-8 numbered balls, which resulted in 64 possible pairs. What if we had 1-1000 numbered balls. Then it will be tedious to carry out by sample space and expected value way. How do we estimate parameter in such cases.