Does the law of large numbers hold for a large number of different trials?

Question

The Law of Large numbers states that when a large number of repeated trials have been completed, the average of the obtained results will be close to the expected value.

However, consider a large number of different trials. Different meaning that there are many of only one trial each.

Here's an example, say that we have 1 million people flip a coin only once each. So only one trial from each person, which totals to 1 million trials. Would the average of the observed values still be 50/50? Does the law still hold for different trials?

Answer 1

No there is no difference between one person flipping a million coins and a million people flipping one coin because at the end of all the flipping you're looking for the same statistic of the same data. They aren't really different trials because when computing the average you're no longer treating them as separate.

Answer 2

Assuming the conditions of the coin tosses are the same, there is no difference between one "trial" of one million flips and one million "trials" of one flip, as @Hugh notes.

However, that assumption seems to throw away any distinction between trials, if I understand correctly with your notion of "different".

If we define a trial to be the one coin flip under a set of conditions C... (C may include any conditions which affect the toss like wind speed, velocity of the toss, height, etc...) than your question is specifically asking if the Law of Large Numbers irons away the differences between a million different sets of conditions so that we arrive at 50/50 the same way as if we did one million flips under one set of conditions.

In other words, does the LoLN hold for one million trials under C in the same way as one trial each under C_1, C_2, ... C_1000000?

The answer is yes. Scientists do this all the time. In an ideal world we would be able to control all the conditions surrounding a trial, and just isolate the variable we want to study. However, that is impossible (though sometimes we are more able to do so than others). The next best thing is to obtain a large sample and study that data, in the hopes that we will iron out the differences through many trials.