Is the law of large numbers accidental?
The law of large numbers is inevitable in the mathematical world. However, the application of the law of large numbers in the real world is of course different, mainly because the premise required by the law of large numbers cannot be perfectly satisfied in the real world. Sometimes the conditions we provide are far from the premise required by the law of large numbers (for example, the mathematical expectation of random variables is infinite, or random variables are significantly related or non-distributed), so applying the law of large numbers can only lead to absurd results; Sometimes the conditions we provide are so close to the premise of the law of large numbers that we think the difference can be ignored, so the conclusion asserted by the law of large numbers is meaningful to us. Take coin toss as an example: although the front and back sides of the coin may not be exactly the same, although each coin toss (whether done manually or by machine) cannot be truly independent of each other, although the coin may be slightly worn after each coin toss test, although the test environment such as temperature and airflow may change slightly with the test. We can reasonably believe that the prerequisite of the law of large numbers has been well satisfied: coins are the same. Coin toss is a mathematical random experiment, and the result of each coin toss is an independent and identically distributed random variable. At this time, we can believe the conclusion of the law of large numbers: if the front of the coin is recorded as 1 and the back is recorded as 0, and the results after repeated coin toss are added to make an average, then the average value obtained is almost equal to the mathematical expectation of random test of 0.5; This means that we can always get about 50% heads and 50% tails.