独立和可能性的每日实例

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Independence and Probability
::独立性和可行性

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In everyday situations, is a probability where additional information is known. Finding the probability of a team scoring better in the next match as they have a former olympian for a coach is a conditional probability compared to the probability when a random player is hired as a coach. The additional information of the coach being a former olympian changes the probability being calculated. If the additional information does not ultimately change the probability, then the two events are independent.
::在日常情况下, 是一个已知更多信息的概率。 在下次比赛中, 发现一个球队因拥有前教练奥林匹亚人而得分的概率是有条件的概率, 与随机选手被聘为教练的概率相比。 教练是前奥林匹亚人的附加信息会改变所计算的概率。 如果附加信息不会最终改变概率, 那么这两个事件是独立的 。

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There are many everyday situations having to do with probabilities. It is important for you to be able to differentiate between a regular probability and a conditional probability. Always read problems carefully in order to be sure that you are interpreting the information correctly.   
::许多日常情况都与概率有关。 您必须能够区分正常概率和有条件概率。 总是仔细阅读问题, 以确保您正确解读信息 。

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Real-World Application: Test Accuracy  
::真实世界应用程序: 测试精确度

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A certain test for allergy is said to be 90% accurate. What does this mean? What does this have to do with conditional probability?
::据说某种过敏测试90%准确。 这意味着什么? 这与条件概率有什么关系?

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You should consider four groups of people: 
::你应该考虑四组人:

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1. People with the allergy who test positive for the allergy ( true positive ). 
   ::具有过敏性的人 测试过敏(正阳性)呈阳性。
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2. People with the allergy who test negative for the allergy ( false negative ).
   ::具有过敏性的人 谁测试负过敏(假阴性)。
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3. People without the allergy who test positive for the allergy ( false positive ).
   ::没有过敏的人 测试过敏呈阳性(假阳性)
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4. People without the allergy who test negative for the allergy ( true negative ). 
   ::没有过敏的人 谁测试负过敏(其实是负的)。

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If a test is 90% accurate, it implies that:
::如果测试准确度达到90%,则意味着:

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1. If a person has the allergy, 90% of the time they will receive a positive test result.  $\begin{array}{r}P(\text{positive}|\text{allergy})=90\mathrm{\%}\end{array}$  
   ::如果一个人过敏,90%的时间将获得阳性测试结果。 P(阳性)过敏=90%
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2. If a person does not have the allergy, 90% of the time they will receive a negative test result.  $\begin{array}{r}P(\text{negative}|\text{no allergy})=90\mathrm{\%}\end{array}$  
   ::如果一个人没有过敏,90%的时间将获得负试验结果。 P(负过敏)=90%

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The 90% is a conditional probability in each case. Note that these are two completely different probability calculations, and they do not automatically have to be the same. It is in fact more realistic if these two probabilities are different. 
::90%是每种情况下的有条件概率。 请注意, 这些是两种完全不同的概率计算, 它们不一定非得是同一的 。 事实上, 如果这两种概率不同, 则更现实 。

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Real-World Application: Spam Emails 
::Real- World 应用程序: 垃圾邮件邮件

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10% of the emails that Michelle receives are spam emails. Her spam filter catches spam 95% of the time. Her spam filter misidentifies non-spam as spam 2% of the time. Let  $\begin{array}{r}A\end{array}$ be the event that an email is spam . Let  $\begin{array}{r}B\end{array}$ be the event that the spam filter identifies the email as spam .
::米歇尔收到的电子邮件中有10%是垃圾邮件。 她的垃圾邮件过滤器在95%的时间里捕捉垃圾邮件。 她的垃圾邮件过滤器误认为非垃圾邮件在2%的时间里是垃圾邮件。 如果电子邮件在垃圾邮件中出现, 请让B成为垃圾邮件过滤器将电子邮件识别为垃圾邮件的事件 。

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1. What does  $\begin{array}{r}P(A)\end{array}$ mean in English?
::1. P(A)在英语中意味着什么?

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$\begin{array}{r}P(A)\end{array}$ is the probability that a random email is spam.
::P(A) 是随机电子邮件是垃圾邮件的概率 。

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2. What does  $\begin{array}{r}P(B|A)\end{array}$ mean in English?
::2. P(BA)在英语中意味着什么?

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$\begin{array}{r}P(B|A)\end{array}$ is the probability that a spam email gets identified as spam.
::P(BA) 是垃圾邮件被识别为垃圾邮件的概率 。

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3. What does  $\begin{array}{r}P(B^{\mathrm{\prime }}|A^{\mathrm{\prime }})\end{array}$ mean in English?
::3. P(BA})在英语中意味着什么?

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$\begin{array}{r}P(B^{\mathrm{\prime }}|A^{\mathrm{\prime }})\end{array}$ is the probability that a non-spam email does not get identified as spam .
::P(BA}) 是非垃圾邮件不被识别为垃圾邮件的概率 。

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4. Find $\begin{array}{r}P(A).\end{array}$
::4. 查找P(A)。

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$\begin{array}{r}P(A)=10\mathrm{\%}\end{array}$
::P(A)=10%

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5. Find  $\begin{array}{r}P(B|A).\end{array}$
::5. 查找P(BA)。

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$\begin{array}{r}P(B|A)=95\mathrm{\%}\end{array}$
::P(BA)=95%

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6. Find $\begin{array}{r}P(B^{\mathrm{\prime }}|A^{\mathrm{\prime }}).\end{array}$
::6. 查找P(BA})。

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$\begin{array}{r}P(B^{\mathrm{\prime }}|A^{\mathrm{\prime }})=98\mathrm{\%}.\end{array}$  Note that in the problem, 2% is  $\begin{array}{r}P(B|A^{\mathrm{\prime }}).P(B|A^{\mathrm{\prime }})\end{array}$ and $\begin{array}{r}P(B^{\mathrm{\prime }}|A^{\mathrm{\prime }})\end{array}$  must add to 100% because $\begin{array}{r}B\end{array}$  and $\begin{array}{r}{B}^{\mathrm{\prime }}\end{array}$  are complements.
::P(BA})=98%。请注意,在问题中,2%是P(BA}),P(BA})和P(BA})必须增加100%,因为B和B是补充性的。

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Examples
::实例实例实例实例

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Example 1
::例1

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10% of the emails that Michelle receives are spam emails. Her spam filter catches spam 95% of the time. Her spam filter misidentifies non-spam as spam 2% of the time. What percent of the emails in the spam folder are not spam emails?
::Michelle收到的电子邮件中有10%是垃圾邮件。她的垃圾邮件过滤器在95%的时间里捕捉垃圾邮件。她的垃圾邮件过滤器误认为非垃圾邮件在2%的时间里是垃圾邮件。垃圾邮件文件夹中的电子邮件中有多少不是垃圾邮件?

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This question is asking for the probability that an email that has been identified as spam is a regular email, $\begin{array}{r}P(A^{\mathrm{\prime }}|B)\end{array}$ . You were not given this probability directly. One way to approach this problem is to make a two-way frequency table for a some number of emails. Suppose you have 1000 emails.
::这个问题是问一个被确认为垃圾邮件的电子邮件是普通电子邮件的概率, P(AB) 。 您没有直接获得这种概率 。 解决这一问题的一个方法就是为一些电子邮件制作双向频表。 假设您有1000个电子邮件 。

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You know that 10% of those (100 emails) will be spam . This means 90% of those (900 emails) will not be spam . Fill these numbers into the table.
::你知道其中的10% (100个电子邮件) 将是垃圾邮件。 这意味着其中的90% (900个电子邮件) 将不会是垃圾邮件。 将这些数字填入表格 。

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You also know that 95% of the spam emails (95 emails) will be identified as spam . This means the other 5 spam emails will not be identified as spam . Fill these numbers into the table.
::您也知道,95%的垃圾邮件( 95个邮件) 将被识别为垃圾邮件。 这意味着其他5个垃圾邮件不会被识别为垃圾邮件。 将这些数字填入表格 。

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You also know that 98% of the non-spam emails (882 emails) will not be identified as spam . This means that the other 18 emails will be identified as spam . Fill these numbers into the table.
::您也知道98%的非垃圾邮件( 882 个电子邮件) 将不会被识别为垃圾邮件。 这意味着其他 18 个电子邮件将被识别为垃圾邮件。 将这些数字填入表格 。

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Now go back to the question. The question is asking for the probability that an email that has been identified as spam is a regular email. 113 emails that were identified as spam. 18 of them are not spam emails.  $\begin{array}{r}P(A^{\mathrm{\prime }}|B)=\frac{18}{113}\approx 16\mathrm{\%}.\end{array}$  Even though the spam filter is pretty accurate, 16% of the emails in the spam folder will be regular emails.
::现在回到问题所在。 问题是要问一个被确认为垃圾邮件的电子邮件是常规电子邮件的可能性。 113个被确认为垃圾邮件的电子邮件。 其中18个不是垃圾邮件。 P( AB) = 18113 16%。 尽管垃圾邮件过滤器相当准确, 垃圾邮件文件夹中的16%的电子邮件将是常规电子邮件 。

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Example 2
::例2

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Karl takes the bus to school. Each day, there is a 10% chance that his bus will be late, a 20% chance that he will be late, and a 2% chance that both he and the bus will be late. Let  $\begin{array}{r}C\end{array}$ be the event that Karl is late . Let  $\begin{array}{r}D\end{array}$ be the event that the bus is late .
::卡尔每天坐公交车去上学。每天,他的公交车会晚到的可能性是10%,他晚到的可能性是20%,他和公交车都会晚到的可能性是2%。让C代表卡尔迟到的事件。让D代表晚到的事件。

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i) State the 10%, 20%, and 2% probabilities in probability notation in terms of events  $\begin{array}{r}C\end{array}$ and $\begin{array}{r}D.\end{array}$
:一) 说明C和D事件概率符号的10%、20%和2%的概率概率。

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$\begin{array}{rl}10\mathrm{\%}& =P(D).\\ 20\mathrm{\%}& =P(C).\\ 2\mathrm{\%}& =P(C\cap D).\end{array}$
::10P(D).20P(C).2P(C)D。

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ii) Are events  $\begin{array}{r}C\end{array}$ and  $\begin{array}{r}D\end{array}$ independent? Explain.
:二) C和D事件是否独立?

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Events  $\begin{array}{r}C\end{array}$ and  $\begin{array}{r}D\end{array}$ are independent if $\begin{array}{r}P(C\cap D)=P(C)P(D).\end{array}$
::如果P(CD)=P(C)P(D),事件C和D是独立的。

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$\begin{array}{rl}P(C)P(D)& =(0.2)(0.1)\\ & =0.02\\ & =2\mathrm{\%}\\ & =P(C\cap D)\end{array}$
::P(C)P(D)=(0.2)(0.1)=0.02=2P(CD)

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Therefore, the events are independent.
::因此,这些事件是独立的。

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iii) Find the probability that Karl is not late but the bus is late.
:三) 找到Karl没有迟到但公交车迟到的概率。

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This is $\begin{array}{r}P(C^{\mathrm{\prime }}\cap D).\end{array}$  Because the two events are independent, $\begin{array}{r}P(C^{\mathrm{\prime }}\cap D)=P(C^{\mathrm{\prime }})P(D).\end{array}$  Since there is a 20% chance that Karl will be late, there is an 80% chance that Karl will not be late. This means $\begin{array}{r}P(C^{\mathrm{\prime }})=80\mathrm{\%}.\end{array}$  Therefore,
::这是P(CD) 。 因为这两个事件是独立的, P(CD) = P(CD) = P(C) P(D) 。 由于Karl 迟到的可能性为20%, Karl 晚到的可能性为80%。 这意味着 P(CD) = 80%。 因此,

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$\begin{array}{rl}P(C^{\mathrm{\prime }}\cap D)& =P(C^{\mathrm{\prime }})P(D)\\ & =(0.8)(0.1)\\ & =8\mathrm{\%}.\end{array}$
::P(CD)=P(C)P(D)=(0.8)(0.1)=8%。

Summary

播放段落 Conditional probability is when the probability of a second event is affected by the probability of the first event. The formula for conditional probability “A given B”: $$\begin{array}{r}P(A|B)=\frac{P(A\cap B)}{P(B)}\end{array}$$ ::有条件概率是当第二个事件的概率受第一个事件的概率影响时。 有条件概率“ 给定 B” 的公式: P( AB) = P( AB) P( B) 播放段落 Two events are independent if one event occurring does not change the probability of the second event occurring.  $$\begin{array}{r}P(A\cap B)=P(A)P(B)\end{array}$$ ::如果发生一个事件并不改变发生第二个事件的概率,则两个事件是独立的。 P(AB)=P(A)P(B)

---

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Review
::审查审查审查审查

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0.1% of the population is said to have a new disease. A test is developed to test for the disease. 97% of people without the disease will receive a negative test result. 99.5% of people with the disease will receive a positive test result. Let  $\begin{array}{r}D\end{array}$ be the event that a random person has the disease. Let  $\begin{array}{r}E\end{array}$ be the event that a random person gets a positive test result.
::据说,0.1%的人口患有新疾病。为检测该疾病,开发了测试。97%的无该疾病者将获得负面测试结果。99.5%的患者将获得正测试结果。如果是随机患者患上该疾病,请将D列为随机患者。如果是随机患者,则E为随机患者获得正测试结果。

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1. State the 0.1%, 97%, and 99.5% probabilities in probability notation in terms of events  $\begin{array}{r}D\end{array}$ and $\begin{array}{r}E\end{array}$ .
::1. 以事件D和E的概率符号表示0.1%、97%和99.5%的概率概率。

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Fill in the two-way frequency table for this scenario for a group of 1,000,000 people. Follow the steps to help.
::填入这一假设情景的双向频率表,供100万人使用,并按步骤提供帮助。

	播放段落 Disease ::疾病 疾病	播放段落 No Disease ::无疾病	播放段落 Total ::共计共计共计共计
播放段落 Positive Test ::阳性试验
播放段落 Negative Test ::负负试验
播放段落 Total ::共计共计共计共计			1,000,000

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2. How many of the 1,000,000 people have the disease? How many don't have the disease?
::2. 1 000 000人中有多少人患有这种疾病?有多少人没有这种疾病?

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3. How many of the people without the disease will receive a negative test result (true negative)? How many of the people without the disease will receive a positive test result (false positive)?
::3. 有多少没有感染该疾病的人将获得负面检测结果(实际上为负)?有多少没有感染该疾病的人将获得正测试结果(假正)?

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4. How many of the people with the disease will receive a positive test result (true positive)? How many of the people with the disease will receive a negative test result (false negative)?
::4. 有多少患有这种疾病的人将获得正测试结果(正数)?有多少患有这种疾病的人将获得负数测试结果(假阴数)?

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5. What does  $\begin{array}{r}P(D|E)\end{array}$ mean in English?
::5. P(DE)在英语中意味着什么?

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6. Find $\begin{array}{r}P(D|E)\end{array}$ . Is this a surprising result?
::6. 找出P(D)-E。这是令人惊讶的结果吗?

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7. What does  $\begin{array}{r}P(D|E{)}^{\mathrm{\prime }}\end{array}$ mean in English?
::7. P(DE) 英文是什么意思?

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8. Find $\begin{array}{r}P(D|E^{\mathrm{\prime }})\end{array}$ .
::8. 查找P(DE)。

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9. Are the two events  $\begin{array}{r}D\end{array}$ and  $\begin{array}{r}E\end{array}$ independent? Justify your answer.
::9. 两个事件D和E是否独立?

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After finishing his homework, Matt often plays video games and/or has a snack. There is a 60% chance that Matt plays video games, an 80% chance that Matt has a snack, and a 55% chance that Matt plays video games and has a snack. Let  $\begin{array}{r}G\end{array}$ be the event that Matt plays video games and  $\begin{array}{r}S\end{array}$ be the event that Matt has a snack.
::完成作业后,马特经常玩电子游戏和/或吃零食。 马特有60%的机会玩电子游戏,马特有80%的机会吃零食,马特有55%的机会玩电子游戏和吃零食。 让G成为马特玩电子游戏和S的事件,马特也有零食。

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10. State the 60%, 80%, and 55% probabilities in probability notation in terms of events $\begin{array}{r}G\end{array}$ and $\begin{array}{r}S\end{array}$ .
::10. 说明G和S事件概率表示的概率概率为60%、80%和55%。

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11. Are events  $\begin{array}{r}G\end{array}$ and  $\begin{array}{r}S\end{array}$ independent? Explain.
::11. G和S事件是否独立?

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12. Consider 100 days after Matt has finished his homework. Use the probabilities in the problem to fill in the two-way frequency table.
::12. 考虑一下Matt完成作业后100天的情况,利用问题的概率填补双向频率表。

	播放段落 Snack ::草纸	播放段落 No Snack ::无草纸	播放段落 Total ::共计共计共计共计
播放段落 Video Games ::视频游戏游戏
播放段落 No Video Games ::无视频游戏
播放段落 Total ::共计共计共计共计			100

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13. Given that Matt played video games, find the probability that he had a snack.
::13. 鉴于Matt在玩电子游戏,找到他有零食的可能性。

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14. What is $\begin{array}{r}P(G|S)\end{array}$  in English?
::14. 英文的P(G)Q(S)是什么?

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15. Find $\begin{array}{r}P(G|S)\end{array}$ .
::15. 查找P(GS)。

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16. A dollar-bill changer on a snack machine was tested with 100 $1 bills. Twenty-five of the bills were found to be counterfeit, but only one was accepted by the machine. However six of the legal bills were rejected. Draw a chart to show the number of legal and counterfeit bills that were accepted or rejected.
::16. 在零食机上,用100美元的钞票测试了一个美元钞票兑换机,发现其中25项是伪造的,但只有1项被机器接受,但有6项被否决,绘制一张图表,显示被接受或拒绝的法律钞票和假钞的数量。

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1. What is the probability that a bill will be rejected given that it is legal?
   ::鉴于一项法案是合法的,其被否决的可能性有多大?
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2. What is the probability that the counterfeit bill is accepted?
   ::假钞被接受的可能性有多大?

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Review (Answers)
::审查(答复)

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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 。