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统计代写|Naive definition of probability stat 代写

统计代写|Naive definition of probability stat 代写

统计代考

Historically, the earliest definition of the probability of an event was to count the number of ways the event could happen and divide by the total number of possible outcomes for the experiment. We call this the naive definition since it is restrictive and relies on strong assumptions; nevertheless, it is important to understand, and useful when not misused.

Definition 1.3.1 (Naive definition of probability). Let $A$ be an event for an experiment with a finite sample space $S$. The naive probability of $A$ is
$$
P_{\text {naive }}(A)=\frac{|A|}{|S|}=\frac{\text { number of outcomes favorable to } A}{\text { total number of outcomes in } S} .
$$
(We use $|A|$ to denote the size of $A$; see Section A. $1.5$ of the math appendix.) In terms of Pebble World, the naive definition just says that the probability of $A$ is the fraction of pebbles that are in $A$. For example, in Figure $1.1$ it says
$$
P_{\text {naive }}(A)=\frac{5}{9}, P_{\text {naive }}(B)=\frac{4}{9}, P_{\text {naive }}(A \cup B)=\frac{8}{9}, P_{\text {naive }}(A \cap B)=\frac{1}{9} .
$$

For the complements of the events just considered,
$$
P_{\text {naive }}\left(A^{c}\right)=\frac{4}{9}, P_{\text {naive }}\left(B^{c}\right)=\frac{5}{9}, P_{\text {naive }}\left((A \cup B)^{c}\right)=\frac{1}{9}, P_{\text {naive }}\left((A \cap B)^{c}\right)=\frac{8}{9} .
$$
In general,
$$
P_{\text {naive }}\left(A^{c}\right)=\frac{\left|A^{c}\right|}{|S|}=\frac{|S|-|A|}{|S|}=1-\frac{|A|}{|S|}=1-P_{\text {naive }}(A)
$$
In Section 1.6, we will see that this result about complements always holds for probability, even when we go beyond the naive definition. A good strategy when trying to find the probability of an event is to start by thinking about whether it will be easier to find the probability of the event or the probability of its complement. De Morgan’s laws are especially useful in this context, since it may be easier to work with an intersection than a union, or vice versa.

The naive definition is very restrictive in that it requires $S$ to be finite, with equal mass for each pebble. It has often been misapplied by people who assume equally likely outcomes without justification and make arguments to the effect of “either it will happen or it won’t, and we don’t know which, so it’s 50-50”. In addition to sometimes giving absurd probabilities, this type of reasoning isn’t even internally consistent. For example, it would say that the probability of life on Mars is $1 / 2$ (“either there is or there isn’t life there”), but it would also say that the probability of intelligent life on Mars is $1 / 2$, and it is clear intuitively-and by the properties of probability developed in Section $1.6$-that the latter should have strictly lower probability than the former. But there are several important types of problems where the naive definition is applicable:

  • when there is symmetry in the problem that makes outcomes equally likely. It is common to assume that a coin has a $50 \%$ chance of landing Heads when tossed, due to the physical symmetry of the coin. ${ }^{1}$ For a standard, well-shuffled deck of cards, it is reasonable to assume that all orders are equally likely. There aren’t certain overeager cards that especially like to be near the top of the deck; any particular location in the deck is equally likely to house any of the 52 cards.
  • when the outcomes are equally likely by design. For example, consider conducting a survey of $n$ people in a population of $N$ people. A common goal is to obtain a simple random sample, which means that the $n$ people are chosen randomly with all subsets of size $n$ being equally likely. If successful, this ensures that the naive definition is applicable, but in practice this may be hard to accomplish because of various complications, such as not having a complete, accurate list of contact information for everyone in the population.
统计代写|Naive definition of probability

统计代考

从历史上看,事件概率的最早定义是计算事件可能发生的方式的数量,然后除以实验可能结果的总数。我们称其为幼稚定义,因为它具有限制性并且依赖于强假设;尽管如此,理解这一点很重要,并且在不被滥用时很有用。

定义 1.3.1(概率的朴素定义)。假设 $A$ 是有限样本空间 $S$ 的实验的事件。 $A$ 的天真概率是
$$
P_{\text {naive }}(A)=\frac{|A|}{|S|}=\frac{\text { 有利于 } A}{\text { } S 中的结果总数} 。
$$
(我们使用 $|A|$ 来表示 $A$ 的大小;参见数学附录的 A 节。$1.5$。)就 Pebble World 而言,朴素的定义只是说 $A$ 的概率是分数以 $A$ 为单位的鹅卵石。例如,在图 $1.1$ 中它说
$$
P_{\text {naive }}(A)=\frac{5}{9}, P_{\text {naive }}(B)=\frac{4}{9}, P_{\text {naive }} (A \cup B)=\frac{8}{9}, P_{\text {naive }}(A \cap B)=\frac{1}{9} 。
$$

对于刚刚考虑的事件的补充,
$$
P_{\text {naive }}\left(A^{c}\right)=\frac{4}{9}, P_{\text {naive }}\left(B^{c}\right)=\ frac{5}{9}, P_{\text {naive }}\left((A \cup B)^{c}\right)=\frac{1}{9}, P_{\text {naive }} \left((A \cap B)^{c}\right)=\frac{8}{9} 。
$$
一般来说,
$$
P_{\text {naive }}\left(A^{c}\right)=\frac{\left|A^{c}\right|}{|S|}=\frac{|S|-|A |}{|S|}=1-\frac{|A|}{|S|}=1-P_{\text {天真}}(A)
$$
在 1.6 节中,我们将看到这个关于补码的结果总是适用于概率,即使我们超出了朴素的定义。尝试找出事件的概率时,一个好的策略是首先考虑是否更容易找到事件的概率或其补码的概率。德摩根定律在这种情况下特别有用,因为使用交叉口可能比使用联合更容易,反之亦然。

朴素的定义非常严格,因为它要求 $S$ 是有限的,每个卵石的质量相等。人们经常误用它,他们在没有正当理由的情况下假设同样可能的结果,并争论“它会发生或不会发生,我们不知道哪个,所以它是 50-50”。除了有时给出荒谬的概率外,这种推理甚至在内部也不一致。例如,它会说火星上有生命的概率是 $1 / 2$(“要么有生命,要么没有生命”),但它也会说火星上有智慧生命的概率是 $1 / 2 $,并且根据第 1.6 节中开发的概率属性,直观地很清楚,后者的概率应该严格低于前者。但是有几种重要类型的问题适用于幼稚的定义:

  • 当问题中存在使结果同样可能的对称性时。由于硬币的物理对称性,通常假设硬币在被抛掷时有 50 美元 \%$ 的机会正面朝上。 ${ }^{1}$ 对于一副标准的、经过良好洗牌的纸牌,可以合理地假设所有订单的可能性相同。没有某些过度渴望的牌特别喜欢靠近牌组的顶部;牌组中的任何特定位置都有可能放置 52 张牌中的任何一张。
  • 当结果在设计上同样可能时。例如,考虑在 $N$ 人的人口中对 $n$ 人进行调查。一个共同的目标是获得一个简单的随机样本,这意味着 $n$ 人是随机选择的,所有大小为 $n$ 的子集的可能性相同。如果成功,这将确保幼稚的定义适用,但在实践中,由于各种复杂情况,这可能难以实现,例如没有针对人群中每个人的完整、准确的联系信息列表。

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