Expected value without complete sample space

Question

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.

Answer 1

This question mixes up two different things.

Sometimes we can define an entire sample space and all the associated probabilities from the nature of the question. Questions about dice, balls in bags and coins being flipped are like this. Even with 1000 balls in a bag, you can figure out everything. You don't need to write down every possible draw - you can use math. True, the sample space for two balls is 1,000,000, but that doesn't really matter.

Other times we don't have this. Even if you know the entire population, you don't know all the probabilities a priori. If, for example, you are estimating the heights of all the students in a class. You know the entire population but you don't know their heights. Here, you might get the entire population. But suppose you wanted it for an entire university? Tens of thousands of students. It would be close to impossible to get everyone's height, so you take a sample.