随机事件、概率与独立性Random Events, Probability & Independence
怎样用概率描述一次不能预先确定的结果？How can probability describe an outcome that cannot be known in advance?
前置知识：Prerequisites: B01：函数与坐标。B01: functions and coordinates.
复习前置：Review prerequisites: B01 · 函数、坐标与参数Functions, Coordinates & Parameters
学完后你能：After this lesson you can: 列出事件，比较频率和概率，并解释放回与独立性。List events, compare frequency with probability, and explain replacement and independence.
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从装有两红一蓝的盒子抽两次，切换是否放回；比较两次都抽红的理论概率与多次试验频率。Draw twice from a box with two red and one blue ball. Toggle replacement and compare the theoretical probability of two reds with repeated-trial frequency.
从观察到解释From observation to explanation
列出可能结果List possible outcomes
一次抽样的结果可能是红或蓝，事件可以包含多个结果。例如连续两次抽红是一个事件；红球有两颗时，不能把“红、蓝”误当作等可能。One draw can be red or blue, and an event can contain multiple outcomes. Drawing red twice is an event; with two red balls, red and blue are not equally likely.
规则决定概率The rule determines probability
先抽红的概率为 2/3。若放回，第二次仍为 2/3，两次都红是 4/9；若不放回，第一次抽红后剩一红一蓝，第二次抽红为 1/2，两次都红是 1/3。The first red has probability 2/3. With replacement, the second also has probability 2/3 and two reds have probability 4/9. Without replacement, a red first leaves one red and one blue; the chance of two reds is (2/3)(1/2)=1/3.
概率与频率Probability and frequency
理论概率由试验规则给出；频率是实际发生次数除以试验次数。重复试验的频率会波动，次数多时通常更接近理论值，但任何有限次数都不保证完全相等。Theoretical probability follows from the trial rules; frequency is the observed count divided by the number of trials. Frequency fluctuates and often approaches the theoretical value with more trials, but finite trials need not match it exactly.
公式与符号Formula and symbols
P(A) 是事件 A 的概率；N 是试验次数，N_A 是 A 出现次数，p̂ 是观察频率；A∩B 表示 A 与 B 同时发生。独立时联合概率等于各自概率之积。P(A) is the probability of event A; N is the number of trials, N_A the number in which A occurs, and p̂ the observed frequency. A∩B means both events occur. For independent events, the joint probability is the product.
留意这个区别Watch for this
独立表示一个事件发生不改变另一个事件的概率；互斥表示不能同时发生。非零概率的互斥事件不是独立事件。Independence means one event does not change the probability of the other; mutually exclusive events cannot both occur. Mutually exclusive events with nonzero probabilities are not independent.
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