课程： 11.3 有条件概率-interactive

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有条件概率

Consider a high school with 400 students. The two-way table below shows the number of students in each grade who earned different midterm letter grades.

Suppose you choose a student at random from the school. Let

$\begin{align*}G\end{align*}$ be the event that the student is from ninth grade . Let

$\begin{align*}A\end{align*}$ be the event that the student got an

$\begin{align*}A\end{align*}$ . You can calculate the probabilities of each of these events using the information in the table.

$\begin{align*}P(G) &= \frac{20+30+30+15+5}{400}=\frac{100}{400}=\frac{1}{4}=0.25\\ P(A) &= \frac{20+18+25+18}{400} =\frac{81}{400}=0.2025\end{align*}$
::P(G)=20+30+30+30+15+5400=100400=14=0.25P(A)=20+18+25+18400=81400=0.2025

Now suppose you choose a student at random and they tell you they are in 9th grade. What's the probability that they got an $\begin{align*}A?\end{align*}$ Now, your sample space is only the 100 students in 9th grade.
::现在假设你随机选择一个学生,他们告诉你他们是九年级学生。他们获得A的概率是多少?现在,你的样本空间只有九年级的100名学生。

$\begin{align*}P(9\text{th grade student got an } A)\end{align*}$ $\begin{align*}=\frac{20}{100}=0.20.\end{align*}$ When you have additional information that causes you to restrict the sample space you are considering, you are working with conditional probability.
::P( 9 年级学生有 A) = 20100=0. 200. 0. 0. 0. . 当您掌握了导致您限制您正在考虑的样本空间的额外信息时, 您的工作带有有条件的概率。

The conditional probability of event $\begin{align*}A\end{align*}$ given event $\begin{align*}B\end{align*}$ is the probability of event $\begin{align*}A\end{align*}$ occurring given event $\begin{align*}B\end{align*}$ occurred . The notation is $\begin{align*}P(A|B),\end{align*}$ which is read as "the probability of $\begin{align*}A\end{align*}$ given $\begin{align*}B".\end{align*}$
::事件A给定事件B的有条件概率是发生事件A给定事件B的概率。标记为P(AB),读作“A给定事件B的概率”。

With the students example, you were calculating $\begin{align*}P(A|G)\end{align*}$ ("the probability that a student got an $\begin{align*}A\end{align*}$ given that they are in 9th grade"). To calculate this probability, you found the number of ninth grade students with an $\begin{align*}P(A \cap G)\end{align*}$ and divided by the number of ninth grade students $\begin{align*}(G).\end{align*}$ You would have gotten the same result had you found $\begin{align*}P(A \cap G)\end{align*}$ and divided by $\begin{align*}P(G),\end{align*}$ because the 400's cancel each other out.
::以学生为例, 您正在计算 P (AG) (“ 学生在九年级获得A的概率 ” ) 。为了计算这个概率, 您发现九年级学生中持有P (AG) 的人数, 并除以九年级学生中学生的人数。如果您找到P (AG) 并被P(G) 分开, 您将会得到同样的结果, 因为四百个学生相互取消。

$\begin{align*}P(A|G)=\frac{20}{100}=\frac{\frac{20}{400}}{\frac{100}{400}}=\frac{P(A \cap G)}{P(G)}\end{align*}$
::P(AG)=20100=20000000=P(AG)P(G)

To calculate a conditional probability, you can always use this formula, generalized below:
::要计算一个条件概率,您可以总是使用以下通用的公式:

$\begin{align*}P(A|B)=\frac{P(A \cap B)}{P(B)}; \ P(B) \neq 0\end{align*}$
::P(AB) = P(AB) P(B); P(B) 0

Interactive Die Roll
::交互式Die滚滚

Use the interactive below to explore the probability of rolling two even numbers in a row on a six-sided die.
::使用下面的交互式互动来探索在六面死亡中一行滚动两个偶数的概率。

Let's take a look at some problems involving conditional probability
::让我们来研究一下有条件概率的一些问题

1. Consider the experiment of tossing three coins and recording the sequence of heads and tails. Let $\begin{align*}A\end{align*}$ be the event of getting at least two heads . Let $\begin{align*}B\end{align*}$ be the event of getting three heads .
::1. 考虑将三个硬币扔掉并记录头和尾的顺序的实验,让一个事件至少能吸引两个头,让B事件能吸引三个头。

Find $\begin{align*}P(A|B).\end{align*}$
::查找 P( AB) 。

The sample space for tossing three coins is $\begin{align*}S = \{ HHH, HHT, HTH, THH, HTT, THT, TTH, TTT \}.\end{align*}$ $\begin{align*}P(A|B)\end{align*}$ is the probability of getting at least two heads given that you have gotten three heads. If you KNOW that you got three heads, then you automatically have gotten at least two heads. $\begin{align*}P(A|B)\end{align*}$ should be 1. Using the formula:
::扔三个硬币的样本空间是 SHHH、HHT、HTH、THH、THT、TTT、TTT}.P(AB) 是至少得到两个头的概率,因为你有三个头。如果你知道你有三个头,那么你就会自动得到至少两个头。 P(AB)应该是 1 。使用公式:

$\begin{align*}P(A|B)=\frac{P(A \cap B)}{P(B)}=\frac{\frac{1}{8}}{\frac{1}{8}}=1\end{align*}$
::P(AB) = P(AB) P(B) = 1818= 1

Find $\begin{align*}P(B|A).\end{align*}$
::查找 P( BA) 。

$\begin{align*}P(B|A)\end{align*}$ is the probability of getting three heads given that you have gotten at least two heads. Since you know you have gotten at least two heads , the new sample space is $\begin{align*}\{HHT,HTH,THH,HHH\}.\end{align*}$ The probability of getting three heads is $\begin{align*}\frac{1}{4}.\end{align*}$ Using the formula:
::P(BA) 是指获得三个头的概率,因为你至少已经得到了两个头。因为你知道你至少得到了两个头,所以新的样本空间是 {HHT,HTH,THH,HHH}。获得三个头的概率是 14。使用公式:

$\begin{align*}P(B|A)=\frac{P(B \cap A)}{P(A)}=\frac{\frac{1}{8}}{\frac{4}{8}}=\frac{1}{4}\end{align*}$
::P(BA)=P(BA)P(A)=1848=14

Does $\begin{align*}P(A|B)=P(B|A)?\end{align*}$
::P(AB)=P(BA)?

$\begin{align*}P(A|B) \neq P(B|A).\end{align*}$ The order of the letters within the probability statement matters!
::P(AB)P(BA) 概率说明中字母的顺序要紧!

The sample space for tossing three coins is $\begin{align*}S=\{HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\}.\end{align*}$
::扔三个硬币的样本空间是SHHHHHHHHHHTHTHTHTTTTT}。

2. Consider two $\begin{align*}C\end{align*}$ and $\begin{align*}D.\end{align*}$ What is $\begin{align*}P(C|D)\end{align*}$ in terms of $\begin{align*}P(C)\end{align*}$ and $\begin{align*}P(D)?\end{align*}$ What is $\begin{align*}P(D|C)\end{align*}$ in terms of $\begin{align*}P(C)\end{align*}$ and $\begin{align*}P(D)?\end{align*}$
::2. 就P(C)和P(D)而言,P(C)和P(D)是什么?就P(C)和P(D)而言,P(D)是什么?

$\begin{align*}P(C|D)\end{align*}$ and $\begin{align*}P(D|C)\end{align*}$ respectively, in terms of $\begin{align*}P(C)\end{align*}$ and $\begin{align*}P(D)\end{align*}$ are .
::就P(C)和P(D)分别而言,P(C)和P(D)分别为P(C)和P(D)。

3. You have two coins, one regular coin and one special coin with heads on both sides. You put the two coins in a bag and choose one at random. Let $\begin{align*}S\end{align*}$ be the event that the coin is the special coin with heads on both sides. Let $\begin{align*}H\end{align*}$ be the event that when the coin is tossed it comes up heads .
::3. 你有两个硬币,一个普通硬币,一个两边头的硬币,你把两个硬币放在一个袋子里,随意选择一个。让S来说明硬币是两边头的特别硬币。让H来说明硬币被扔出头顶的时候。

What is $\begin{align*}P(S)?\end{align*}$
::什么是P(S)?

$\begin{align*}P(S)\end{align*}$ is the probability that you have chosen the special coin. There are two coins in the bag, one of which is the special coin. $\begin{align*}P(S)=\frac{1}{2}.\end{align*}$
::P(S) 是您选择特殊硬币的概率。袋子里有两枚硬币, 其中一枚是特殊硬币。 P(S)=12 。

What is $\begin{align*}P(S|H)?\end{align*}$
::什么是P(S)H?

The experiment of choosing a coin and tossing it has four outcomes in its sample space:
::选择硬币和抛掷硬币的实验在其样本空间有四个结果:

Regular Coin, Tails
::普通硬币, 反面

Regular Coin, Heads
::普通硬币, 头部

Special Coin, Heads 1st Side
::特别币首首首方

Special Coin, Heads 2nd Side
::特种部队,第二排,第二排

$\begin{align*}P(S|H)\end{align*}$ is the probability that you have chosen the special coin given that when you tossed it, it came up heads. In order to compute this probability, you need to know $\begin{align*}P(S \cap H)\end{align*}$ and $\begin{align*}P(H).\end{align*}$ There are two outcomes that are special coins and heads , so $\begin{align*}P(S \cap H)=\frac{2}{4}=\frac{1}{2}.\end{align*}$ There are three outcomes that are heads , so $\begin{align*}P(H)=\frac{3}{4}.\end{align*}$ Now you can compute the conditional probability:
::P(SH) 是您选择特殊硬币的概率, 因为您在抛掷时, 选择了特殊硬币时, 它会抬头。要计算此概率, 您需要知道 P(SH) 和 P(H) 。有两个结果是特殊硬币和头, 所以 P(SH) = 24= 12。有三个结果是头, 所以P(H) = 34。现在您可以计算条件概率 :

$\begin{align*}P(S|H)=\frac{P(S \cap H)}{P(H)}=\frac{\frac{1}{2}}{\frac{3}{4}}=\frac{2}{3}\end{align*}$
::P(SH) = P(SH) P(H) = 1234= 23

Compare the answers. In each case you are calculating the probability that the coin is the special coin; however, in the second part you have additional information that supports that it is the special coin . Because you have additional information, the probability that it is the special coin goes up. Note that because $\begin{align*}P(S) \neq P(S|H),\end{align*}$ the two events $\begin{align*}S\end{align*}$ and $\begin{align*}H\end{align*}$ are NOT independent.
::比较答案。在每种情况下, 您都在计算硬币是特殊硬币的概率; 但是, 在第二部分, 您有额外信息支持它是特殊硬币。由于您有额外信息, 特殊硬币的概率会上升。请注意, 因为 P( S) P( SH) , S 和 H 事件是非独立的。

Examples
::实例实例实例实例

Example 1
::例1

Describe the two ways that you can use probabilities to check whether or not two events are independent.
::描述您可以使用两种方法的概率来检查两个事件是否独立。

Two events $\begin{align*}A\end{align*}$ and $\begin{align*}B\end{align*}$ are independent if and only if:
::两个事件A和B是独立的,如果而且只有在以下情况下:

1) $\begin{align*}P(A \cap B)=P(A) P(B)\end{align*}$
::1) P(AB)=P(A)P(B)

2) $\begin{align*}P(A|B)=P(A)\end{align*}$ and $\begin{align*}P(B|A)=P(B)\end{align*}$
::2) P(AB)=P(A)和P(BA)=P(B)

Example 2
::例2

Consider the experiment of rolling a pair of dice. Let $\begin{align*}A\end{align*}$ be the event that the numbers on the dice have a sum of 8 . Let $\begin{align*}B\end{align*}$ be the event that the two numbers on the dice are a 3 and a 5 .
::想象一下滚动一对骰子的实验。让A来说明骰子上的数字总和是8。让B来说明骰子上的两个数字是3和5。

a) What is $\begin{align*}P(A)?\end{align*}$ What is $\begin{align*}P(B)?\end{align*}$
:a) 什么是P(A)?P(B)是什么?

The sample space for this experiment has 36 outcomes:
::这项实验的样本空间有36个结果:

$\begin{align*}S &= \{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),\\ & (3,1),(3,2),(3,3),(3,4),(3,5),(3,6), (4,1),(4,2),(4,3),(4,4),(4,5),(4,6), \\ & (5,1),(5,2),(5,3),(5,4),(5,5),(5,6), (6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\} \end{align*}$
::S[1,1,1,2,2,2,5,5,5,1,6,2,2,2,2(2、3,2,4,2,5,2,6,6,3(3,3,3,3,3,3,3,3,3,4(3,4,5,3,6,6,6,6,6,6,6,6,6,5,5,5,5,6,6,6,6,5,5,5,6,6,6,6,5,6,6,6,6,6,6,5,6,6,6,6,5,6,6,6,6,6,6,6,7,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,4,6,6,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4

There are 5 pairs of numbers that have a sum of 8, so $\begin{align*}P(A)=\frac{5}{36}.\end{align*}$ There are 2 pairs of numbers that are a 3 and a 5, so $\begin{align*}P(B)=\frac{2}{36}=\frac{1}{18}.\end{align*}$
::有5对数字,总和为8,所以P(A)=536。有2对数字是3和5,P(B)=236=118。

b) What is $\begin{align*}P(B|A)?\end{align*}$
:b) 什么是P(BA)?

$\begin{align*}P(B|A)=\frac{P(B \cap A)}{P(A)}=\frac{\frac{2}{36}}{\frac{5}{36}}=\frac{2}{5}.\end{align*}$ The sample space has been restricted to the five outcomes with a sum of 8.
::P(BA) = P(BA) P(A) = 236536= 25。样本空间限于五个结果,共8个结果。

c) Are events $\begin{align*}A\end{align*}$ and $\begin{align*}B\end{align*}$ independent?
:c) A和B事件是否独立?

The two events are independent if $\begin{align*}P(B|A)=P(B).\end{align*}$
::如果P(BA)=P(B),这两个事件是独立的。

$\begin{align*}P(B)=\frac{1}{18},\end{align*}$ but $\begin{align*}P(B|A)=\frac{2}{5}. \end{align*}$
::P(B)=118,但P(B)=25。

$\begin{align*}P(B) \neq P(B|A),\end{align*}$ so the events are not independent.
::P(B)P(B)P(B)A),所以事件并不独立。

CK-12 PLIX Interactive: Conditional Probability
::CK-12 PLIX 互动:有条件概率

Summary

Two events are independent if one event occurring does not change the probability of the second event occurring.

$\begin{align*}P(A\cap B)=P(A)P(B)\end{align*}$
::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)

Dependent events are events where one outcome impacts the probability of the other.
::依附事件是指一种结果影响另一种结果概率的事件。

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”:

$\begin{align*}P(A|B)=\frac{P(A\cap B)}{P(B)}\end{align*}$
::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)

Review
::审查审查审查审查

Consider the experiment of drawing a card from a standard deck. Let $\begin{align*}A\end{align*}$ be the event that the card is a diamond. Let $\begin{align*}B\end{align*}$ be the event that the card is a red card. Let $\begin{align*}D\end{align*}$ be the event that the card is a four.
::考虑从标准甲板上绘制一张牌的实验。请将牌是钻石的情况列入 A 。请将牌是红卡的情况列入 B 。请将牌是红卡的情况列入 D 。请将牌是 4 的情况列入 D 。

1. Find $\begin{align*}P(A), P(B), P(C), P(D)\end{align*}$ .
::1. 查找P(A)、P(B)、P(C)、P(D)。

2. Find $\begin{align*}P(A|B)\end{align*}$ and $\begin{align*}P(B|A)\end{align*}$ .
::2. 查找P(AB)和P(BA)。

3. Are events $\begin{align*}A\end{align*}$ and $\begin{align*}B\end{align*}$ independent?
::3. A和B事件是否独立?

4. Find $\begin{align*}P(D|B)\end{align*}$ and $\begin{align*}P(B|D)\end{align*}$ .
::4. 查找P(DB)和P(BD)。

5. Are events $\begin{align*}B\end{align*}$ and $\begin{align*}D\end{align*}$ independent?
::5. B和D事件是否独立?

Consider the experiment of flipping three coins and recording the sequence of heads and tails. Let $\begin{align*}A\end{align*}$ be the event that all the coins are the same. Let $\begin{align*}B\end{align*}$ be the event that there is at least one heads. Let $\begin{align*}C\end{align*}$ be the event that the third coin is a tails. Let $\begin{align*}D\end{align*}$ be the event that the first coin is a heads.
::想象一下翻三个硬币的实验, 并记录头和尾的顺序。让A来记录所有硬币都相同的事件。让B来记录至少有一个头的事件。让C来记录第三个硬币是尾巴的事件。让D来记录第一个硬币是头的事件。

6. Find $\begin{align*}P(A), P(B), P(C), P(D)\end{align*}$ .
::6. 查找P(A)、P(B)、P(C)、P(D)。

7. Find $\begin{align*}P(A|B)\end{align*}$ and $\begin{align*}P(B|A)\end{align*}$ .
::7. 查找P(AB)和P(BA)。

8. Are events $\begin{align*}A\end{align*}$ and $\begin{align*}B\end{align*}$ independent?
::8. A和B事件是否独立?

9. Find $\begin{align*}P(C|D)\end{align*}$ and $\begin{align*}P(D|C)\end{align*}$ .
::9. 查找P(CD)和P(DC)。

10. Are events $\begin{align*}C\end{align*}$ and $\begin{align*}D\end{align*}$ independent?
::10. C和D事件是否独立?

11. Find $\begin{align*}P(A|D)\end{align*}$ and $\begin{align*}P(D|A)\end{align*}$ .
::11. 查找P(AD)和P(DA)。

12. Are events $\begin{align*}A\end{align*}$ and $\begin{align*}D\end{align*}$ independent?
::12. A和D事件是否独立?

13. Explain what conditional probability is in your own words.
::13. 解释用你自己的话来说,有条件概率是多少。

14. Explain two ways to test whether or not two events are independent. When does it make sense to use one method over the other?
::14. 解释两种检验两种事件是否独立的方法:使用一种方法比另一种方法合理吗?

15. Explain where the conditional probability formula $\begin{align*}P(A|B)=\frac{P(A \cap B)}{P(B)}\end{align*}$ comes from.
::15. 解释有条件概率公式P(AB)=P(AB)P(B)的来源。

16. In the city there are 3 major clothing stores: B, G, and L. 60% of the population shops at store B, 36% shop at store G, and 34% shop at store L. You also know that 18% shop at both stores B and L, 15% shop at both B and G, 4% shop at G and L, and 2% shop at all three locations. 5% of the population does not shop at any of these three stores. (Note, be careful of overlapping percentages!)
::16. 城市有3家主要服装商店:B、G和L.60%的商店B商店、G商店36%的商店和L商店34%的商店。你也知道B和L商店18%的商店、B和L商店15%的商店、G和L商店4%的商店以及所有三个地点2%的商店。 5%的人口不在这三家商店中的任何一家商店购物。 (注意注意重复的百分比! )

What is the probability that a person will shop at store B given they shop in at least one other clothing store?
::如果一个人至少在其他一家服装店购物,那么在B商店购物的可能性有多大?

What is the probability that a person will shop at either store B or store G or both?
::一个人在B商店或G商店或G商店或两者兼而有之的可能性有多大?

17. From a group of 50 students, those taking Calculus and Chemistry this year were counted. 15 students take Chemistry, 7 take both, 22 take neither. Draw a Venn diagram to represent the data. What is the probability that a student is taking Calculus given they are taking Chemistry?
::17. 从一组50名学生中,计算出今年学微积分和化学的学生。 15名学生选化学,7名学生选化学,7名学生选两个,22名学生选两个,22名学生选两个,绘制文恩图来代表数据。学生选微积分的可能性是多少?

18. There are two candy jars on the shelf. In Jar A you notice that there are 2 pink candies and 5 green candies. In Jar B, there are 4 pink candies and 3 green candies. You roll a die to decide what jar to choose from. You roll a five for Jar A and a one for Jar B. Determine the probability that the candy was chosen from Jar B given it was pink.
::18. 架子上有两个糖果罐,在罐A中,你注意到有2个粉色糖果和5个绿糖果罐;在罐B中,有4个粉色糖果和3个绿糖果罐;在罐B中,你为决定从哪罐罐中选择而滚死;在罐A中,你为罐A提供5个罐,在罐B中,你为罐B提供1个罐。确定从罐B中选择糖果的可能性,因为糖果是粉色的。

Review (Answers)
::审查(答复)

To see the answer key for this book, go to the and click on the Answer Key under the ' ' option.
::要查看本书的答案键, 请在“ ” 选项下点击答案键。

章节大纲

常规

有条件概率

Interactive Die Roll ::交互式Die滚滚

Examples ::实例实例实例实例

Example 1 ::例1

Example 2 ::例2

CK-12 PLIX Interactive: Conditional Probability ::CK-12 PLIX 互动:有条件概率

Review ::审查审查审查审查

Review (Answers) ::审查(答复)