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# 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.

## 统计代考

$$P_{\text {naive }}(A)=\frac{|A|}{|S|}=\frac{\text { 有利于 } A}{\text { } S 中的结果总数} 。$$
（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
$$P_{\text {naive }}(A)=\frac{5}{9}, P_{\text {naive }}(B)=\frac{4}{9}, P_{\text {naive }} (A \cup B)=\frac{8}{9}, P_{\text {naive }}(A \cap B)=\frac{1}{9} 。$$

$$P_{\text {naive }}\left(A^{c}\right)=\frac{4}{9}, P_{\text {naive }}\left(B^{c}\right)=\ frac{5}{9}, P_{\text {naive }}\left((A \cup B)^{c}\right)=\frac{1}{9}, P_{\text {naive }} \left((A \cap B)^{c}\right)=\frac{8}{9} 。$$

$$P_{\text {naive }}\left(A^{c}\right)=\frac{\left|A^{c}\right|}{|S|}=\frac{|S|-|A |}{|S|}=1-\frac{|A|}{|S|}=1-P_{\text {天真}}(A)$$

• 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。