In the high school probability chapter, we learned about the concept of randomized experiments. The concepts related to randomized trials are further discussed below.

　　Since the results of random phenomena cannot be predicted in advance, they may seem irregular at first glance. However, people have found that when the same random phenomenon occurs repeatedly in large numbers, the frequency of each possible outcome is stable, indicating that random phenomena also have inherent regularities. The quantitative regularity shown by random phenomena when they appear repeatedly in large numbers is called the statistical regularity of random phenomena . Probability theory and mathematical statistics are a discipline that studies the statistical regularity of random phenomena.
　　Historically, the most famous experiment to study the statistical regularity of random phenomena is the coin tossing experiment. The table below lists some records of coin tossing experiments throughout history.

　　Character introduction: DeMorgan , Buffon , Pearson .

　　* Mathematical experiments

　　Click [ Coin Tossing Experiment ] to conduct a coin tossing simulation test.

　　The experiment shows that although it is impossible to accurately predict in advance whether heads or tails will appear in each toss of a coin, when a large number of repeated experiments are carried out, it is found that the number of heads and tails is approximately equal, that is, the proportion of each to the total number of trials is approximately , and as the experiments As the number of times increases, this ratio tends to more stably. This shows that although the results of random phenomena have no regularity in a few experiments or observations, it can be seen through long-term observation or a large number of repeated experiments that the results of the experiments are regular. This kind of regularity is a random experiment. The characteristics of the result itself.
　　To study the statistical regularity of random phenomena, we need to conduct repeated observations of random phenomena. We call the observation of random phenomena experiments .
　　 For example , observing a shooter's shooting at a fixed target; tossing a coin three times and observing the number of heads it appears; recording the number of calls received by a city's emergency hotline throughout the day and night are all experiments. They have the following common characteristics:
　　(1)  Repeatability: the test can be repeated under the same conditions;
　　(2)  Observability: each test has more than one possible outcome, and all possible results of the test can be clarified in advance;
　　(3)  Uncertainty: The result of each test cannot be accurately predicted in advance, but it is certain that one of the above possible results will occur.
　　In probability theory, we call an experiment with the above three characteristics a random experiment, denoted as .