One: Probability of Random Events (1) Necessary EventsUnder condition S, the event that must happen is called certain event relative to condition S, which is abbreviated as inevitable event. (2) Impossible eventsUnder condition S, events that will never happen are called impossible events relative to condition S, or impossible events for short. (3) Identifying eventsNecessary and impossible events are collectively referred to as deterministic events relative to condition S. (4) Random eventsThe event that may or may not occur under condition S is called random event relative to condition S, which is called random event for short. Determined events and random events are collectively referred to as events, using A, B, C,...Express. (5) Frequency and frequencyRepeat n trials under the same condition S to observe whether an event A occurs. Na is the frequency of event A in n trials; FN (A) is the ratio of event A to relative frequency of event A; for a given random event A, if the frequency of event A increases with the number of trials, FN (A) is stable at a constant.This constant is denoted as P(A), which is called probability of event A. (6) Differences and Relations between Frequency and ProbabilityFrequency of random events refers to the ratio of the number of occurrences of random events to the total number of experiments n. It has a certain stability. It always oscillates near a constant. With the increasing number of experiments, the oscillation range becomes smaller and smaller. We call this constant the probability of random events. The probability reflects the probability of occurrence of random events quantitatively. Frequency is repeated in a large number of trials.The probability of this event can be approximated on the premise of the test. Frequency is an approximate value of probability. As the number of experiments increases, frequency will approach probability more and more. In practical problems, the probability of events is unknown, and frequency is often used as its estimate. Frequency itself is random and can't be determined before the experiment. Repeated experiments with the same number of times will result in different frequencies of events. Probability is a definite number, which exists objectively and has nothing to do with each experiment. For example, if a coin is uniform in texture, the probability of coin tossing up front is 0.5, which is independent of how many experiments are done. Example 1In order to estimate the tail number of fish in the reservoir, the following methods can be used: first, a certain number of fish can be caught from the reservoir, such as 2,000 fish, marked each fish, without affecting its survival, and then put back into the reservoir. After appropriate time, let it mix with the rest of the fish in the reservoir, and then a certain number of fish can be caught from the reservoir, such as 500 fish. Look at the marked fish, there are 40. Based on the above data, the tail number of fish in the reservoir is estimated. Analysis:Students think first, then exchange and discuss, and teachers guide. This is actually a probabilistic problem. That is, the percentage of 2,000 fish in the reservoir accounts for all fish, especially 40 fish with marks in 500 fish. This shows that the probabilities of marking in catching a certain number of fish are solvable. Solution:If the tail number of the fish in the reservoir is n, A={the marked fish}, then P(A)=. Because of P(A), 2 From (1) and (2), the solution is n_25 000. So it is estimated that there are about 25,000 fish in the reservoir. Second: The Significance of Probability 1.Probability is a description of the probability of random events. The greater the probability of random events, the smaller the probability of random events.It is also possible that events with high probability will not occur, and events with low probability will also occur. For example:1If the probability of winning a lottery is zero, will 1,000 tickets win the lottery? 2"The weather forecast said that the probability of precipitation yesterday was 90%. As a result, it didn't rain at all, and the weather forecast was too inaccurate." After learning the probability, can you give an explanation? answer1. It is not necessarily possible to win the lottery, because buying 1,000 lottery tickets is equivalent to doing 1,000 trials, because the results of each trial are random, that is, each lottery ticket may or may not win the lottery. Therefore, there
Summary of three required knowledge points of mathematics in senior high school and elaboration of examples
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