Sampling is not something people just came up with; it has a long mathematical history that firmly grounds it in reality. That history is often unexpected: much of the basis lay in devising formulas that determined the odds of various gambles so that you knew when you were getting fair odds on a bet.
Hundreds of types of statistics are known and used in multitudes of ways. They work in reality. We know that because our entire economy essentially rests on those statistical tests. With trillions of dollars riding on them, they’d better work.
In the OP’s example, all the apples in all the barrels are collectively called the universe. Sampling is used when it is too expensive or time-consuming to examine every example in a universe. Testers are very aware of this, and the relationship between a sample and its universe is part of what drives the variety of statistical tests. Not all universes contain essentially identical elements and so different procedures must be used.
Getting good samples in the real world is an art. I’ve done polling in the field and by computerized dialing. That’s hard, really, really hard. People are hell to deal with. Things are infinitely easier.
But the short answer to the OP is that any objection you can come up with has been looked at by a million people over hundreds of years. Statistics are the daily jobs of enormous hordes. Since the universe of a horde is made of people, not all the results will be correct. Nevertheless, the basic principals are as sound as in any other part of math.
All excellent info being provided here. Apart from the academic and technical discussions I think one other important thing the lay-person should take away from this is the practical difficulties of applying the theory in a real world situation.
A classroom example will often be neat, ring-fenced and with a definitive answer applying to data that you can trust. The real world is rarely like that.
It is messy, it is incomplete, it has unknown history and subject to unknown confounding variables and influences.
Hence the need to couch any conclusions in terms of probabilities, margins of error, how “powerful” tests are, etc. etc. and that part of this subject can be very difficult to accept. We humans love certainty,
We do. Except when betting. Somehow I find that most people can quickly get their heads around “the odds” of sporting outcomes when it comes time to bet on outcomes. Betting on the horses sees punters managing probabilities with some intuitive ease. People still have biases, and people have problems with magnitudes of likelihoods (like buying lottery tickets) but placing a few dollars at odds of 2:1 is quickly grasped.
It is making the transition from that to other statistics where there is still a speed hump.
When I was a high school student, one day we started on basic counting theory and calculating odds in mathematics class. The rather curious outcome was that a few of my cohort were inspired to start up a card school. They took it remarkably seriously, and it ran for at least a month, maybe longer.
In my own UK context it just meant an informal group of people playing cards together. In my local pub we’d often have a 3 card brag or poker “school”.
Yes there is no assumption that it is for teaching purposes although to be honest I did learn many important lessons. Such as “don’t play cards against my brother”
One of Heinlein’s characters said that he financed his education by working as a math tutor. Specifically, he taught probability. Such as teaching suckers not to draw to an inside straight.
Somewhere I read that someone with modern knowledge of probability would clean up at gambling before the modern era (assuming he didn’t get murdered by his debtors I suppose).
This. If your sample size is large enough and representative enough, you can draw some pretty solid conclusions about something. If not, then you run into all sorts of bias and skewed results.
May I recommend an introductory statistics class? They go into all this in some detail in those classes.
For sure a modern with modern math could identify where bad payoff odds are offered on something concrete. E.g. a 10:1 payoff on rolling a 6-sided die that’s as fair as then-current tech can create.
But, e.g., horse racing has always been a true gamble. You can’t count on the long term to rescue you. “Good” horses run badly and “bad” horses run well. As they say in modern professional sports, any given team can beat any other given team on any given day.
Excepting the legendary “Dutch book”, there’s no way better math produces a reliably winning handicapping of e.g. 15th Century horse racing any more than it will at Santa Anita tomorrow.
I think the contest was somehat more subtle things like being able to calculate the odds on something like chuck-a-luck which a naive person would think is fair (but in fact has a definite house advantage)
And even that house advantage isn’t enough for a lot of operators any more. Nowadays, they usually replace the dice with a wheel, where each space has three dice faces marked on it. And if you count them up, those spaces on the wheel have about two or three times as many doubles as you’d expect from fair dice (doubles being where the house makes its money).
