Section: 依赖事件 | 12.18 附属事件

Section outline

Antonio is responsible for washing the dishes. There are four bowls, three plates, and six cups in the sink. If Antonio washes a bowl first, what is the probability that he will grab another bowl next?
::安东尼奥负责洗碗,水槽里有四个碗、三个盘子和六个杯子。如果安东尼奥先洗一个碗,他接下来会抢另一个碗的几率是多少?

In this concept, you will learn how to calculate the probability of two dependent events occurring.
::在此概念中,您将学会如何计算发生两次依附事件的概率。

Calculating Probabilities of Two Dependent Events
::计算两个依附事件的可能性

The Probability Rule is the probability that two , $\begin{aligned} A \end{aligned}$ and $\begin{aligned} B \end{aligned}$ , will both occur and is written as the formula:
$\begin{aligned} P (A and B) = P (A) \cdot P (B) \end{aligned}$

::概率规则是两个A和B的概率,两者同时发生,并写成公式:P(A和B)=P(A)P(B)

If you pull two socks from a bag full of the following blue and red socks, what is the probability of both socks being red?
::如果你从一个袋子里拉两双袜子满满的是下面的蓝色和红色袜子那么两双袜子红的几率是多少?

The probability of the first sock being red is:
::第一只袜子红的概率是:

$\begin{aligned} P (red 1st sock) = \frac{3}{6} = \frac{1}{2} \end{aligned}$

::P(红1号袜)=36=12

What about the second sock? Having removed the first sock from the bag, we have changed the number of red socks and the total number of socks in the bag. So now instead of there being 3 red socks out of 6 total socks, there are only 2 red socks left out of 5 total socks:
::第二只袜子呢?我们把第一只袜子从袋子里拿掉之后,改变了红袜子的数量和袋内袜子的总数。所以现在,在总共6只袜子中,没有3只红袜子,只有5只袜子中的2只红袜子:

$\begin{aligned} P (red 2nd sock) = \frac{2}{5} \end{aligned}$

::P(红2号袜)=25

The probability of two red socks being pulled is:
::两只红袜子被拉动的概率是:

$\begin{aligned} P (red 1st sock and red 2nd sock) & = P (red 1st sock) \cdot P (red 2nd sock) \\ = \frac{1}{2} \cdot \frac{2}{5} \\ = \frac{2}{10} \\ = \frac{1}{5} \end{aligned}$

::P(红1号袜和红2号袜)=P(红1号袜)=P(红2号袜)=P(红2号袜)=1225=210=15

Let's consider another example.
::让我们考虑另一个例子。

A stack of 8 cards has 4 Jacks and 4 Queens. What is the probability of picking 2 Jacks from the stack at random?
::8张牌堆有4张杰克和4张皇后。

Use the Probability Rule to find the probability of the two dependent events.
::使用概率规则来发现两种依赖性事件的概率。

First, determine the probability of the 1st Jack:
::首先,确定第一杰克的概率:

$\begin{aligned} P (1st Jack) = \frac{5}{8} \end{aligned}$

::P(第1杰克)=58

Next, determine the probability of the 2nd Jack:
::接下来,确定第二杰克的概率:

Once the 1st Jack is taken, the probability of 2nd Jack is only 3 of 7 because there are only 4 Jacks left out of 7 total cards:
::一旦第一杰克被拿走,第二杰克的概率只有7分之3, 因为总共7张卡片中只有4张杰克被遗漏了:

$\begin{aligned} P (2nd Jack) = \frac{4}{7} \end{aligned}$

::P(第2杰克)=47

Then, substitute the values into the Probability Rule formula:
::然后,将数值替换为概率公式规则公式:

$\begin{aligned} P (1st Jack and 2nd Jack) & = P (1st Jack) \cdot P (2nd Jack) \\ = \frac{5}{8} \cdot \frac{4}{7} \\ = \frac{5}{14} \end{aligned}$

::P(第1杰克和第2杰克)=P(第1杰克)=P(第1杰克)=P(第2杰克)=5847=514

The answer is the probability of picking two Jacks is $\begin{aligned} \frac{5}{14} \end{aligned}$ .
::答案是选择两个杰克的概率是514

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Antonio, who was washing the dishes.
::之前有人给你提过安东尼奥的问题他洗盘子时在洗盘子

What is the probability that he will wash a bowl first and then grab another bowl next when there are four bowls, three plates, and six cups in the sink?
::当水槽里有四个碗、三个盘子和六个杯子时,他首先洗碗,然后再抢下一个碗的可能性有多大?

First, figure out the total number of dishes in the sink: $\begin{aligned} 4 + 3 + 6 = 13 \end{aligned}$
::首先,找出水槽中的盘子总数:4+3+6=13

Next, write a ratio to show the probability of picking the first bowl and then the second bowl:
::接下来写一个比例来显示选择第一个碗和第二个碗的概率:

probability of picking first bowl = $\begin{aligned} \frac{4}{13} \end{aligned}$
::选择第一碗的概率=413

probability of picking second bowl = $\begin{aligned} \frac{3}{12} = \frac{1}{4} \end{aligned}$
::选择第二个碗的概率= 312=14

Then, multiply the two values together: $\begin{aligned} \frac{4}{13} \times \frac{1}{4} \end{aligned}$ = $\begin{aligned} \frac{4}{52} = \frac{1}{13} \end{aligned}$
::然后,将两个数值相乘: 413x14= 452=113

The answer is the probability of Antonio choosing two bowls is $\begin{aligned} \frac{1}{13} \end{aligned}$ .
::答案是安东尼奥选择两碗的概率是113

Example 2
::例2

There are five girls and eight boys in a group. Mrs. Marsh is going to choose two students randomly to lead the line. What is the probability that she will choose two boys?
::Marsh夫人会随机挑选两个学生担任领导。她选择两个男孩的可能性有多大?

First, figure out the total number of students:
::首先,找出学生总数:

$\begin{aligned} 5 + 8 = 13 \end{aligned}$

Next, write a ratio to show the probability of her picking the first boy and then the second boy:
::接下来写一个比例来显示她选择第一个男孩和第二个男孩的概率:

probability of picking first boy = $\begin{aligned} \frac{8}{13} \end{aligned}$
::选择第一个男孩的概率 = 813

probability of picking second boy = $\begin{aligned} \frac{7}{12} \end{aligned}$
::选择第二个男孩的概率=712

Then, multiply the two values together:
::然后,将两个值乘以一起:

$\begin{aligned} \frac{8}{13} \times \frac{7}{12} = \frac{14}{39} \end{aligned}$

The answer is the probability of Mrs. Marsh choosing two boys is $\begin{aligned} \frac{14}{39} \end{aligned}$ .
::答案是Marsh女士选择两个男孩的概率是1439

A bag has three red marbles, two yellow marbles and four blue marbles.
::一个袋子有三个红色弹珠,两个黄色弹珠和四个蓝色弹珠。

Example 3
::例3

What is the probability of pulling two red marbles out of the bag?
::从袋子里拉出两个红色弹珠的概率是多少?

First, figure out the total number of marbles: $\begin{aligned} 3 + 2 + 4 = 9 \end{aligned}$
::首先,找出弹珠总数:3+2+4=9

Next, write a ratio to show the probability of pulling the first red marble and the second red marble out of the bag:
::接下来,写一个比率,以显示从包中拉出第一个红大理石和第二个红大理石的概率:

the ratio of the probability of pulling the first red marble: $\begin{aligned} \frac{3}{9} = \frac{1}{3} \end{aligned}$
::绘制第一个红大理石的概率比率:39=13

the ratio of the probability of pulling the second red marble: $\begin{aligned} \frac{2}{8} = \frac{1}{4} \end{aligned}$
::拉拉第二个红大理石的概率比率: 28=14

Then, multiply the two values together:
::然后,将两个值乘以一起:

$\begin{aligned} \frac{1}{3} \times \frac{1}{4} \end{aligned}$ = $\begin{aligned} \frac{1}{12} \end{aligned}$

The answer is the probability of choosing two red marbles is $\begin{aligned} \frac{1}{12} \end{aligned}$ .
::答案是选择两个红色弹珠的概率是112。

Example 4
::例4

What is the probability of pulling out two blue marbles?
::拔出两颗蓝色弹珠的概率是多少?

First, figure out the total number of marbles: $\begin{aligned} 3 + 2 + 4 = 9 \end{aligned}$
::首先,找出弹珠总数:3+2+4=9

Next, write a ratio to show the probability of pulling the first blue marble and the second blue marble out of the bag:
::接下来,写一个比率,以显示从包中拉出第一个蓝色大理石和第二个蓝色大理石的概率:

the ratio of the probability of pulling the first blue marble: $\begin{aligned} \frac{4}{9} \end{aligned}$
::绘制第一个蓝色大理石的概率比: 49

the ratio of the probability of pulling the second blue marble: $\begin{aligned} \frac{3}{8} \end{aligned}$
::推拉第二颗蓝色大理石的概率比率:38

Then, multiply the two values together: $\begin{aligned} \frac{4}{9} x \frac{3}{8} = \frac{12}{72} = \frac{1}{6} \end{aligned}$
::然后,将两个数值相乘: 49x38=1272=16

The answer is the probability of choosing two blue marbles is $\begin{aligned} \frac{1}{6} \end{aligned}$ .
::答案是选择两个蓝色弹珠的概率是16

Example 5
::例5

What is the probability of pulling out two yellow marbles?
::拔出两个黄色弹珠的概率是多少?

First, figure out the total number of marbles: $\begin{aligned} 3 + 2 + 4 = 9 \end{aligned}$
::首先,找出弹珠总数:3+2+4=9

Next, write a ratio to show the probability of pulling the first yellow marble and the second yellow marble out of the bag:
::接下来,写一个比率,以显示从包中拉出第一个黄色大理石和第二个黄色大理石的概率:

the ratio of the probability of pulling the first yellow marble: $\begin{aligned} \frac{2}{9} \end{aligned}$
::拉动第一个黄色大理石球概率比: 29

the ratio of the probability of pulling the second yellow marble: $\begin{aligned} \frac{1}{8} \end{aligned}$
::拉拉第二个黄色大理石大理石的概率比率:18

Then, multiply the two values together: $\begin{aligned} \frac{2}{9} \times \frac{1}{8} \end{aligned}$ = $\begin{aligned} \frac{2}{72} = \frac{1}{36} \end{aligned}$
::然后,将两个值乘以:29x18=272=136。

The answer is the probability of choosing two yellow marbles is $\begin{aligned} \frac{1}{36} \end{aligned}$ .
::答案是选择两个黄色弹珠的概率是136

Review
::回顾

Use this description to figure out the probability of each dependent event.
::使用此描述来计算每个依附事件的概率。

A box has eight kittens in it. Three calico, two white and three black.
::一个盒子里有八只小猫,三只卡利科,两只白色和三只黑色

What is the probability of choosing two white kittens?
::选择两只白猫的概率是多少?

What is the probability of choosing three black kittens?
::选择三只黑猫的概率是多少?

What is the probability of choosing two black kittens?
::选择两只黑猫的概率是多少?

What is the probability of choosing two calico kittens?
::选择两只小猫的概率是多少?

What is the probability of choosing three calico kittens?
::选择三只小猫的概率是多少?

What is the probability of choosing one white and then one black kitten?
::选择一只白猫和一只黑猫的概率是多少?

What is the probability of choosing two calico and one black?
::选择两个卡利科和一个黑色的概率是多少?

What is the probability of choosing one calico and one white kitten?
::选择一只卡利科和一只白猫的概率是多少?

What is the probability of choosing a striped kitten?
::选择条纹小猫的概率是多少?

What is the probability of choosing one white kitten and two black kittens?
::选择一只白猫和两只黑猫的概率是多少?

Solve each problem.
::解决每一个问题。

A clothes dryer contains 5 black socks and 1 white sock. What is the probability of taking two socks, one after another, out of the dryer and having them both be black?
::烘干机装有5只黑袜子和1只白袜子。将两只袜子一个接一个地从烘干机上取出来,同时将两只袜子都换成黑色的可能性有多大?

A clothes dryer contains 4 black socks and 2 white socks. What is the probability of taking two socks out of the dryer and having them both be black?
::烘干机内装有4个黑袜子和2个白袜子。从烘干机内取出两只袜子,让两只袜子都变黑的可能性有多大?

A clothes dryer contains 3 black socks and 3 white socks. What is the probability of taking two socks out of the dryer and having them both be black?
::烘干机内装有3个黑袜子和3个白袜子。从烘干机内取出2个袜子,让这两只袜子都变黑的可能性有多大?

A clothes dryer contains 3 black socks and 3 white socks. What is the probability of taking two socks out of the dryer and having the first one be black and the second one be white?
::烘干机内装有3个黑袜子和3个白袜子。从烘干机内取出2个袜子,第一个是黑袜子,第二个是白袜子的可能性有多大?

Bob bought two theater box tickets. The computer randomly assigns the tickets in one of 5 seats: end seat A, middle seat B, middle seat C, middle seat D, or end seat E. What is the probability that the first ticket is A and the second ticket is seat B?
::Bob买了两张剧院票箱票。电脑随机分配了五个座位之一的票:最后座位A,中间座位B,中间座位C,中间座位D,或者最后座位E。第一张票是A,第二张票是B的概率是多少?

Review (Answers)
::回顾(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

Calculating Probabilities of Two Dependent Events ::计算两个依附事件的可能性

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Calculating Probabilities of Two Dependent Events
::计算两个依附事件的可能性

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)