I’ve been surfing a lot of physics forums lately, and I’ve run across several “misguided” interpretations of the law of large numbers. For example:
The law of large numbers is rigorously provable from the axioms pf probability.
What it says is if a trial (experiment or whatever) is repeated a large number of times, independently under identical conditions, then the proportion of times that any specified outcome occurs approximately equals the probability of the event’s occurrence on any particular trial; the larger the number of repetitions, the better the approximation tends to be.
This guarantees a sufficiently large, but finite, number of trials exists (ie an ensemble) that for all practical purposes contains the outcomes in proportion to its probability.
You have to be very careful with statements like this. If you take the “sufficiently large, but finite, number of trials” to be unbounded (that is, you don’t associate any number N with it in advance, but rather wait until you get something close to the expected probability, then yes, the law of large numbers does say that.
But if you pick N in advance (no matter how large), the expected result is not a guarantee. It’s an expected feature of random behavior that any deviation from randomness, no matter how unlikely, will happen eventually. Take tossing a fair coin, for example. If you flip it forever, you will find runs of N heads, or N tails, for any (even very large) value of N. The bigger N is the less often that happens, but it does happen, and the correct interpretation of the law of large numbers tells you exactly how often, on average.
So if you try to pick N in advance, then there is some (very small) probability that you will get a significant deviation from the expected behavior over your N trials. There is no guarantee of anything – just likelihoods that are perhaps very much in your favor.
This doesn’t mean you can’t reason using this sort of experimentation. If you pick up a coin and flip it 100 times and get 100 heads, then it’s very, very likely that the coin is unfair. It’s not unreasonable at all for you to proceed with your work on that presumption. But there is no guarantee the coin is unfair. While it’s unlikely your particular 100 flips just happened to capture one of the unusual behavior sequences, it might have.
When discussing the subtle nuances of science and math, it’s important to get this sort of thing right. Probability theory is an incredibly powerful tool, but it’s important to understand it for what it is.