Course: 11.4 双体频率表-interactive

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双体频率表

Suppose you conduct a survey where you ask each person two questions. Once you have finished conducting the survey, you will have two pieces of data from each person. Whenever you have two pieces of data from each person, you can organize the data into a two-way frequency table .
::假设您在调查中询问每个人两个问题。完成调查后, 您将拥有来自每个人的两块数据。当您从每个人那里获得两块数据时, 您可以将数据组织成双向频率表。

A group of people were surveyed about 1) whether they have cable TV and 2) whether they went on a vacation in the past year .
::对一群人进行了调查1) 他们是否有有线电视;(2) 他们去年是否度假。

The numbers in the frequency table show the numbers of people that fit each pair of preferences. For example, 97 people have cable TV and took a vacation last year . 38 people have cable TV but did not take a vacation last year . The totals of the rows and columns have been added to the frequency table for convenience. From the far right column you can see that 135 people have cable TV and 31 people don't have cable TV for a total of 166 people surveyed. From the bottom row you can see that 111 people took a vacation and 55 people did not take a vacation for a total of 166 people surveyed.
::频率表中的数字显示了适合每种偏好的人数。例如,97人有有线电视,去年休假。 38人有有有线电视,但去年没有休假。为了方便起见,在频率表中增加了行和列的总数。从最右侧的栏目中可以看到,135人有有有线电视,31人没有有线电视,总共166人接受了调查。从底层的栏目中可以看到,111人度假,55人没有休假,总共166人接受了调查。

You can use the two-way frequency table to calculate probabilities about the people surveyed. For example, you could find:
::您可以使用双向频率表计算被调查者的概率。例如,您可以找到:

The probability that a random person selected from this group took a vacation last year.
::从这个组中挑选出来的随机人员去年休假的概率。

The probability that a random person from this group who has cable TV took a vacation last year.
::去年有有有线电视的这个团体的随机人员去度假的概率。

Whether or not “choosing a person with cable TV” and “choosing a person who took a vacation” are for this population of 166 people.
::是否“选择使用有线电视的人”和“选择休假的人”是属于这166人的人口。

Let's look at some problems involving two way frequency tables.
::让我们来研究一下涉及两个频率表的一些问题。

1. Suppose you choose a person at random from the group surveyed below. Let $\begin{aligned} A \end{aligned}$ be the event that the person chosen took a vacation last year . Find $\begin{aligned} P (A) . \end{aligned}$
::1. 假设您从下面调查的人群中随机选择一个人。请将所选择的人去年休假的情况作为例。查找 P( A) 。

There were 166 people surveyed, so there are 166 outcomes in the sample space. 111 people took a vacation last year.
::调查了166人,抽样空间有166个结果。去年有111人休假。

$\begin{aligned} P (A) = \frac{111}{166} \approx 0.67 or 67 % \end{aligned}$
::P(A)=11111660.67或67%

INTERACTIVE

Frequency Table

Over the sample space of the total amount of people surveyed, people were categorized within Event A or its complement Not A as well as Event B or its complement Not B. Since these events are mutually exclusive, everyone surveyed must have fallen into one of the four boxes dividing the different combinations of A + B, ~A + B, ~A + ~B, and A + ~B.
::在所调查的总人数的抽样空间中,人们被归类为事件A或其补充活动非A和补充活动B或补充活动B或补充活动非B。由于这些事件是相互排斥的,因此接受调查的每个人都必须掉入A+B、~A+B、~A+B、~A+~B和A+~B等不同组合的四个方框之一。

In the frequency graph below, you can adjust the amount of people who fall into these categories using the up and down arrows in the four boxes. For Event C, you can adjust is description with its buttons such that the box on the table representing Event C is highlighted and the probability of C over the total people surveyed can be calculated below.
::在下面的频率图中,您可以使用四个框中的向上箭头和向下箭头来调整属于这些类别的人数。对于事件C,您可以调整用其按钮来描述,以便突出显示表上代表事件C的框,下面可以计算出C相对于被调查总人数的概率。

2. Suppose you choose a person at random from the group surveyed below. Let $\begin{aligned} A \end{aligned}$ be the event that the person chosen took a vacation last year . Let $\begin{aligned} B \end{aligned}$ be the event that the person chosen has cable TV . Find $\begin{aligned} P (A | B) . \end{aligned}$
::2. 假设您从下面调查的组别中随机选择一个人。请将选中的人去年休假的事件作为A。请将选中的人有有有线电视的事件作为B。找到 P( AB ) 。

You are looking for the probability that the person took a vacation given that they have cable TV. Since you know that the person has cable TV, the sample space has been restricted to the 135 people with cable TV . 97 of those people took a vacation .
::您正在寻找一个人休假的概率,因为他们有有有线电视。因为您知道这个人有有有线电视, 样本空间仅限于有有有线电视的135人。其中97人度假。

$\begin{aligned} P (A | B) = \frac{97}{135} \approx .72 or 72 % \end{aligned}$
::P(AB)=9713572%或72%

Suppose you wanted to use the formula for this calculation.
::假设你想用公式来计算

$\begin{aligned} P (A | B) = \frac{P (A \cap B)}{P (B)} = \frac{\frac{97}{166}}{\frac{135}{166}} = \frac{97}{135} \approx 0.72 or 72 % \end{aligned}$
::P(AB)=P(AB)P(B)=97166135166=971350.72或72%

With the conditional probability formula, each probability is calculated with the sample space of 166. The two 166's cancel each other out, and the result is the same. Sometimes it makes sense to use the conditional probability formula, and sometimes it is easier to think logically about what is being asked.
::使用有条件概率公式,每种概率都用166的样本空间计算。两个166的样本空间相互勾销,结果是一样的。有时使用有条件概率公式是有道理的,有时从逻辑上考虑被询问的问题比较容易。

3. Suppose you choose a person at random from the group surveyed below. Let $\begin{aligned} A \end{aligned}$ be the event that the person chosen took a vacation last year . Let $\begin{aligned} B \end{aligned}$ be the event that the person chosen has cable TV . Are events $\begin{aligned} A \end{aligned}$ and $\begin{aligned} B \end{aligned}$ independent?
::3. 假设您从下面调查的人群中随机选择一个人。请将所选择的人去年休假的事件作为A,将所选择的人有有有线电视的事件作为B。事件A和B是否独立?

From #1, you know that $\begin{aligned} P (A) = 67 % . \end{aligned}$ From #2, you know that $\begin{aligned} P (A | B) = 72 % . \end{aligned}$ Because these probabilities are not equal, the two events are NOT independent (they are dependent). People with cable TV are more likely to have taken a vacation as opposed to people without cable TV, so knowing that a person has cable TV increases the probability that they took a vacation.
::从 # 1 , 您知道 P( A) = 67 % 。从 # 2 , 您知道 P( A) = 72% 。由于这些概率不相等, 这两个事件并不独立( 他们有依附关系 ) 。有有线电视的人更有可能休假, 而不是没有有线电视的人, 所以知道一个人有有有线电视会增加他们休假的概率。

Examples
::实例实例实例实例

Example 1
::例1

You are in charge of choosing the theme for the junior/senior prom. You survey the juniors and seniors and record the results in a two-way frequency table.
::您负责选择初级/高级舞会的主题。您对初级和高级学生进行检查, 并将结果记录在双向频表。

Based on the results, you decide to go with the Masquerade Ball.
::根据结果,你决定参加化妆舞会

a) On prom night, what's the probability that a student chosen at random got the theme of his/her choice?
:a) 在舞会之夜,随机挑选的学生获得其选择主题的概率是多少?

This question includes the entire population of 749 people as the sample space . 371 of those people wanted the Masquerade Ball .
$\begin{aligned} P (got their choice) = \frac{371}{749} \approx .50 or 50 % . \end{aligned}$
::这个问题包括所有749人作为抽样空间。这些人中有371人想要化妆舞会。 P(自己选择)=371749.50或50%。

b) What's the probability that a senior chosen at random got the theme of his/her choice?
:b) 随机挑选的高级人选获得其选择主题的可能性有多大?

This is a conditional probability question, because you know that the student is a senior . You can restrict the sample space to just the 387 seniors. 159 of them wanted the Masquerade Ball .
$\begin{aligned} P (senior got their choice) = \frac{159}{387} \approx 41 % . \end{aligned}$
::这是一个有条件的概率问题, 因为您知道学生是高年级学生。您可以将样本空间限制在387名高年级学生。 159人想要化妆舞会。 P( 高级学生) =159387 41% 。

c) What's the probability that a junior chosen at random got the theme of his/her choice?
:c) 随机挑选的初中生获得其选择主题的可能性有多大?

This is a conditional probability question, because you know that the student is a junior . You can restrict the sample space to just the 362 juniors. 212 of them wanted the Masquerade Ball .
$\begin{aligned} P (junior got their choice) = \frac{212}{362} \approx 59 % . \end{aligned}$
::这是一个有条件的概率问题, 因为您知道学生是初级学生。您可以将样本空间限制在362个初级学生。 212人想要化妆舞会。 P( 初级学生) =21236259% 。

Example 2
::例2

A group of 110 students was surveyed about what grade they were in and whether they preferred dogs or cats. 20 9th graders preferred dogs, 5 9th graders preferred cats, 16 10th graders preferred dogs, 4 10th graders preferred cats, 22 11th graders preferred dogs, 6 11th graders preferred cats, 30 12th graders preferred dogs, and 7 12th graders preferred cats.
::一组110名学生接受了调查,了解他们在哪一年级,以及他们喜欢狗还是猫。 20名九年级学生喜欢狗,5名九年级学生喜欢猫,16名十年级学生喜欢狗,4名十年级学生喜欢猫,22名十一年级学生喜欢狗,6名十一年级学生喜欢猫,30名十二年级学生喜欢狗,7名十二年级学生喜欢猫。

a) Construct a two-way frequency table to organize this data.
:a) 建立一个双向频率表来组织这一数据。

b) Suppose a person is chosen at random from this group. Let $\begin{aligned} C \end{aligned}$ be the event that the student prefers cats . Let $\begin{aligned} T \end{aligned}$ be the event that the student is in 10th grade . Find $\begin{aligned} P (C) \end{aligned}$ and $\begin{aligned} P (C | T) . \end{aligned}$
:b) 假设一个人是随机从这个群体中挑选出来的。让C成为学生喜欢猫的事件。让T成为学生在10年级时的活动。找到P(C)和P(C)Q(T)。

There are 110 students total. 22 of them prefer cats. $\begin{aligned} P (C) = \frac{22}{110} = 20 % . \end{aligned}$ $\begin{aligned} P (C | T) \end{aligned}$ means the probability that the student prefers cats given that they are in 10th grade . Restrict the sample space to the 20 10th grade students. 4 of them prefer cats. $\begin{aligned} P (C | T) = \frac{4}{20} = 20 % . \end{aligned}$
::共有110名学生,其中22人更喜欢猫。P(C)=22110=20%。P(C)=20%。P(C)=T)表示学生更喜欢猫的概率,因为他们在10年级。将样本空间限制在20年级10年级的学生。其中4人更喜欢猫。P(C)=420=20%。

c) Are events $\begin{aligned} C \end{aligned}$ and $\begin{aligned} T \end{aligned}$ independent?
:c) C和T事件是否独立?

The events are independent because $\begin{aligned} P (C) = P (C | T) . \end{aligned}$ Being in 10th grade does not affect the probability of the student preferring cats.
::这些事件是独立的, 因为 P( C) = P( CT) 。进入十年级并不影响学生喜欢猫的概率。

CK-12 PLIX Interactive: Sums and Differences of Independent Random Variables
::CK-12 PLIX 互动: 独立随机变量的汇总和差异

Summary

A frequency table is a table that summarizes a data set by stating the number of times each value occurs within the data set.
::频率表是一个表格,它通过说明数据集内每个数值发生次数来汇总一组数据。

Random is probability means everything in a population has an equal chance of being selected.
::随机是概率意味着所有人口都有被选中的同等机会

The formula for finding the conditional probability: $\begin{aligned} P (A | B) = \frac{P (A \cap B)}{P (B)} \end{aligned}$
::查找条件概率的公式: P(AB) = P(AB) P(B)

Review
::审查审查审查审查

Use the following information for #1-#5: A group of 64 people were surveyed about the type of movies they prefer. 12 females preferred romantic comedies, 10 females preferred action movies, and 3 females preferred horror movies. 8 males preferred romantic comedies, 25 males preferred action movies, and 6 males preferred horror movies.
::为#1-#5使用以下信息:一组64人接受了调查,了解他们喜欢的电影类型。 12名女性喜欢浪漫喜剧,10名女性喜欢行动电影,3名女性喜欢恐怖电影。 8名男性喜欢浪漫喜剧,25名男性喜欢行动电影,6名男性喜欢恐怖电影。

1. Construct a two-way frequency table to organize this data.
::1. 构造一个双向频率表以组织此数据。

Suppose a person is chosen at random from this group.
::假设一个人是从这一组中随机挑选出来的。

2. Let $\begin{aligned} F \end{aligned}$ be the event that the person is female. Find $\begin{aligned} P (F) \end{aligned}$ .
::2. 让F成为该人是女性的事件,找到P(F).

3. Let $\begin{aligned} R \end{aligned}$ be the event that the person prefers romantic comedies. Find $\begin{aligned} P (R) \end{aligned}$ .
::3. 让R成为人们喜欢浪漫喜剧的事件。

4. Find $\begin{aligned} P (F | R) \end{aligned}$ and $\begin{aligned} P (R | F) \end{aligned}$ . Explain how these two calculations are different.
::4. 找出P(FR)和P(RF),解释这两个计算方法有何不同。

5. Are events $\begin{aligned} F \end{aligned}$ and $\begin{aligned} R \end{aligned}$ independent? Justify your answer.
::5. F事件和R事件是否独立?

The middle school students in your town were surveyed and classified according to grade level and response to the question “how do you usually get to school”? The data is summarized in the two-way table below.
::根据年级和对“你通常如何上学”问题的答复,对你们城镇的中学生进行了调查和分类?数据摘要见下表双向表格。

	Walk ::步行步行步行	Bus ::公车	Car ::车车车	Total ::共计共计共计共计
6th Grade ::六年级六年级	30	120	65	215
7th Grade ::七年级七年级	25	170	25	220
8th Grade ::8年级八年级	40	130	41	211
Total ::共计共计共计共计	95	420	131	646

6. If a student is chosen at random from this group, what is the probability that he or she is a 6th grade student who takes the bus?
::6. 如果学生是随机从这一群体中挑选出来的,他或她是乘公交车的6年级学生的概率有多大?

7. If a 6th grade student is chosen at random from this group, what is the probability that he or she takes the bus?
::7. 如果从这一群体中随机挑选一个六年级学生,他或她搭乘公共汽车的概率有多大?

8. If a student who takes the bus is chosen at random from this group, what is the probability that he or she is in 6th grade?
::8. 如果乘坐公交车的学生是随机从这一群体中挑选出来的,那么六年级学生的几率有多大?

9. The previous three questions each have to do with 6th grade and taking the bus. Why are the answers to these questions different?
::9. 前三个问题都与六年级和乘公交有关,为什么对这些问题的答案不同?

10. Are the events “being in 6th grade” and “taking the bus” independent? Justify your answer.
::10. " 六年级 " 和 " 搭公车 " 的事件是否独立?

A hospital runs a test to determine whether or not patients have a particular disease. The test is not always accurate. The two-way table below summarizes the numbers of patients in the past year that received each result.
::医院进行检测,以确定病人是否患有某种疾病,检测结果并不总是准确的,下面的双向表格总结了去年每例结果的患者人数。

	Positive Result on Test ::试验阳性结果	Negative Result on Test ::试验负结果	Total ::共计共计共计共计
Has Disease ::有疾病	100	4	104
Does Not Have Disease ::没有疾病	12	560	572
Total ::共计共计共计共计	112	564	676

11. If a patient is chosen at random from this group, what is the probability that he or she has the disease?
::11. 如果病人是随机从这一群体中挑选出来的,他或她患上这一疾病的概率有多大?

12. A patient from this group received a positive test result. What is the probability that he or she has the disease?
::12. 来自这一群体的病人获得了阳性测试结果,他或她患上该疾病的概率有多大?

13. A patient from this group has the disease. What is the probability that he or she received a positive result on the test?
::13. 这一群体的病人患有这种疾病,在测试中获得阳性结果的概率是多少?

14. A “false positive” is when a patient receives a positive result on the test, but does not actually have the disease. What is the probability of a false positive for this sample space?
::14. " 假阳性 " 是指病人在试验中获得正结果,但实际上没有这种疾病。

15. How many of the 676 patients received accurate test results?
::15. 676名病人中有多少人获得了准确的测试结果?

16. In a book club consisting of 50 readers, 36 like detective stories, 20 like science fiction and 12 did not like either. If you select a reader randomly from the group, what is the probability that he or she:
::16. 在一个由50名读者、36名侦探故事、20名科幻小说和12名也不喜欢的书友会中,如果随机从团体中挑选一名读者,那么他或她的概率是:

likes both science fiction and detective stories?
::既喜欢科幻小说也喜欢侦探故事?

likes at least one of science fiction and detective stories?
::喜欢至少一个科幻小说和侦探故事?

likes detective stories given that he or she likes science fiction?
::喜欢侦探故事因为他或她喜欢科幻小说?

likes science fiction given that he or she likes detective stories?
::喜欢科幻小说因为他(她)喜欢侦探故事?

are liking science fiction stories and detective stories independent? Why or why not?
::喜欢科幻小说和侦探小说独立吗?

17. You are given a bag containing 8 red and 5 blue balls. Create a problem that illustrates the concept of probability of drawing at least 2 balls from the bag? Does it make a difference if you replace the balls after each pick? Explain.
::17. 给您一个装有8个红球和5个蓝色球的袋子。产生一个问题, 说明从袋子中至少抽取2个球的概率概念。如果每次选一球后换掉球, 会有区别吗? 解释。

18. A new drug is coming out on the market where it proclaims to grow hair faster than any other alternative. In a sample of 4,000 people, 2,400 were given the drug and of these only 1,600 showed any hair growth. The other 1,600 were given the placebo and 1,200 of these showed hair growth. Draw a two way frequency table to illustrate the data. Determine at least three conclusions you can make from the table.
::18. 市场上出现了一种新的药物,它宣称其头发生长速度比其他任何替代药物都快,在4 000人的抽样中,2 400人服用了这种药物,其中只有1 600人显示出任何头发生长,另外1 600人服用了安慰剂,1 200人表现出了头发增长,绘制了两个频率表来说明数据,确定至少可以从表中得出三个结论。

Review (Answers)
::审查(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Section outline

General

双体频率表

Examples ::实例实例实例实例

Example 1 ::例1

Example 2 ::例2

CK-12 PLIX Interactive: Sums and Differences of Independent Random Variables ::CK-12 PLIX 互动: 独立随机变量的汇总和差异

Review ::审查审查审查审查