The key characteristics of a binomial experiment(also called of Bernoulli trial), are that we have a fixed number of trials, each with two possible outcomes, and the probability of "success" is the same in each trial. We also assume that all the "trials" are independent. 二项分布（又名伯努利实验）的主要特点有，试验次数是固定的，每一次试验都有两个可能的结果，而且每一次“成功”的概率都是相等的。我们同时还假定所有的试验都是各自独立的。

Probability is the study of events and outcomes involving an element of uncertainty. 概率学是一门研究不确定事件和结果的学问。

Probabilities do not tell us what will happen for sure; they tell us what is likely to happen and what is less likely to happen. Sensible people can make use of these kinds of numbers in business and life. 概率并不会确凿地告诉我们将会发生什么，但我们通过概率计算能够知道很有可能发生什么，不太可能发生什么。聪明的人会使用这类数据为自己的事业和生活指明方向。

Probability can also sometimes tell us after the fact what likely happened and what likely did not happened. 概率有时候还可以在事后告诉我们，什么情况是最有可能发生的，而什么情况是最没有可能发生的。

Often it is extremely valuable to know the likelihood of multiple events' happening.很多时候，了解多重事件同时发生的概率是很有价值的。

The probabilty of two independent events' both happening is the product of their respective probabilities. In other words, the probability of Event A happening and Event B happening the the probability of Event A multiplied by the probabiltiy of Event B. 两个独立事件同时发生的概率取决于这两个事件各自的概率，也就是说，事件A与事件B同时发生的概率是这两个事件发生概率的乘积。

the events are independent, meaning that the outcome of one has no effect on the outcome of another.相互独立的事件，也就是说，一个事件的发生及结果对另一个事件不会造成任何影响。

Suppose you are interested in the probability that one event happens or another event happens: outcome A or outcome B (again assuming that they are independent). In this case, the probability of getting A or B consists of the sum fo their individual probabilities: the probability of A plus the probability of B. 假设你对发生这个事件或那个事件的概率感兴趣，也就是出现结果A或结果B的概率（再次假设两个事件是相互独立的），这个概率就是A和B各自的概率之和：A概率+B概率。

Probability also enables us to calculate what might be the most useful tool in all of managerial decision making, particularly finance: expected value.The expected value takes basic probability one step further. The expected value or payoff from some event, say purchasing a lottery ticket, is the sum of all the different outcomes, each weighted by its probability and payoff. 通过概率计算，我们还可以得到在所有管理决策的过程中，尤其是在金融领域是最实用的统计工具：期望值。期望值是基础概率学的升级版。某个事件如买彩票的期望值或收益，实际上就是所有不同结果的和，其中每个结果都是由各自的概率和收益相乘而来。

In fact, the expected value turns out to be extremely powerful because it can tell you whether a particular event is "fair", given its price and expected outcome. 事实上，期望值是一个非常有用的参考数据，通过比较成本投入和期望收益，你就能知道做这件事是不是“值得”。

An important theorem known as the law of large numbers tells us that as the number of trials increases, the average of the outcomes will get closer and closer to its expected value.有一个重要的定律叫作大数定律，即随着试验次数的增多，结果的平均值会越来越接近期望值。

The broader lesson -- and one of the core lessons of personal finance -- is that you should always insure yourself against any adverse contingency that you cannot comfortably afford to withstand. You should skip buying insurance on everything else. 你应该时刻谨记为那些你无法轻松承受的意外上保险，而其他情况就不要浪费钱了， 这是个人理财的核心原则之一。

Expected value can also help us untangle complex decisions that involve many contingencies at different points in time. 有些事情可能会在不同时段出现各种不同的意外状况，在面临这类复杂抉择时，期望值同样能够帮助我们理清思路。

The decision tree maps out each source of uncertainty and the probabilities associated with all possible outcomes. The end of the tree gives us all the possible payoffs and the probability of each. 决策树形图标出了每一个不确定因素的来源，还有所有可能出现的结果及其概率。在树形图的下方，给出了所有回报可能的金额和概率。

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《赤裸裸的统计学》的全部笔记 52篇