12.1 概率的定义

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  概率的定义

  

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  Brian is planning to have his birthday party this year at the local skate park.  He is a little nervous about the weather, fearing his party might get rained out.  He checks out the past weather history for the date of his party.  In the past ten years, there has only been rain in the area of the skate park twice on the date of the party.  He feels more confident, knowing that the probability is high that there will not be rain.  What is the probability of a rain-free birthday at the skate park for Brian's birthday party?
  ::布莱恩计划今年在当地的滑冰公园举行生日派对。 他对天气有点紧张,担心会下雨。 他检查了派对日期的过去天气史。 在过去的十年里,在派对日期,滑冰公园地区只有两次下雨。 他更有信心,知道不会下雨的可能性很大。 在滑冰公园举行布赖恩生日晚会时,没有雨的生日概率有多大?

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  In this concept, you will learn you how to figure out probability by thinking about favorable outcomes and total outcomes . 
  ::在这个概念中,你将学会如何通过思考有利结果和总结果来判断概率。

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  Defining Probability
  ::界定概率

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  Probability is something that you hear about all the time. Anytime you talk about the chances that something will or won’t happen, you are talking about probability. The trick about probability is that it isn’t just about talking. It is also about math. There are mathematical ways of figuring out the likelihood that an event is going to or not going to occur. 
  ::概率是您一直听到的。每当您谈论某事会发生或不会发生的可能性时,您都在谈论概率。关于概率的秘诀是它不仅仅是谈论。它也是关于数学的。有数学的方法可以找出事件是否会发生的可能性。

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  Probability is the likelihood that an event will occur. It is a mathematical way of calculating how likely an event is likely to occur. An event is a result of an experiment or activity that might include such things as:
  ::概率是事件发生的可能性。它是计算事件发生的可能性的一种数学方法。事件是实验或活动的结果,其中可能包括:

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  • flipping a coin
    ::翻硬币
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  • spinning a spinner
    ::旋转旋转旋转器
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  • rolling a number cube
    ::滚动数字立方体
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  • choosing an item from a jar or bag
    ::从罐或袋中选择项目

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  Probability is calculated by looking at the ratio of favorable outcomes to total outcomes in a given situation. In ratio form, the probability of an event is:
  ::概率的计算方法是,在特定情况下,通过查看有利结果与总结果之比来计算有利结果。

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  $$\begin{array}{r}P(\text{event})=\text{favorable outcomes}:\text{total outcomes}\end{array}$$

  
  ::P(活动)=有利成果:总结果

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  An outcome is a possible result of some event occurring. For example, when you flip a coin, “heads” is one outcome; tails is a second outcome. Total outcomes are computed simply by counting all possible outcomes.
  ::某些事件的结果是可能发生的。 例如,当您翻硬币时,“头”是结果之一;尾是第二个结果。总结果仅通过计算所有可能的结果来计算。

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  Keep in mind as you go through this concept that all outcomes used are presumed to be “fair.” For example, when you flip a coin, the outcomes of heads or tails are equally likely. When you spin a spinner, sections are all of equal size and equally likely to be landed on. When you toss a number cube, faces of the cube are the same size and again equally likely to be landed on. And so on.
  ::当您在经历这个概念时,记住所有使用的结果都被假定为“公平 ” 。 比如, 当您翻转硬币时, 头或尾的结果同样可能。 当您旋转一个旋转器时, 区块大小相同, 并同样可能着陆。 当您抛掷一个数字立方体时, 立方体的面部大小相同, 并且同样可能着陆。 等等 。

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  For flipping a coin:
  ::翻硬币:

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  $$\begin{array}{rl}\text{total outcomes}& =\text{heads, tails}\\ & =2\text{total outcomes}\end{array}$$

  
  ::总结果=头部,尾部=2

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  For tossing a dice:
  ::抛骰子用 :

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  $$\begin{array}{rl}\text{total outcomes}& =\cdot 1\cdot \cdot 2\cdot \cdot \cdot 3\cdot \cdot \cdot \cdot 4\cdot \cdot \cdot \cdot \cdot 5\cdot \cdot \cdot \cdot \cdot \cdot 6\\ & =6\text{total outcomes}\end{array}$$

  
  ::6=6 总计结果

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  For selecting a day of the week:
  ::选择一周的一天 :

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  $$\begin{array}{rl}\text{total outcomes}& =\text{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}\\ & =7\text{total outcomes}\end{array}$$

  
  ::星期一、星期二、星期三、星期四、星期五、星期六、星期日=7

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  Favorable outcomes are the specific outcomes you are looking for.
  ::喜人的成果是你们正在寻找的具体成果。

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  For flipping a coin and having it come up heads:
  ::翻硬币 并让硬币浮出头来:

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  $$\begin{array}{rl}\text{favorable outcomes}& =\text{heads, tails}\\ & =1\text{favorable outcome}\end{array}$$

  
  ::有利结果=头部,尾部=1 有利结果

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  For tossing a number cube and having it come with up an even number:
  ::用于抛掷一个数字立方体, 并使它产生一个偶数 :

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  $$\begin{array}{rl}\text{favorable outcomes}& =\cdot 1\cdot \cdot 2\cdot \cdot \cdot 3\cdot \cdot \cdot \cdot 4\cdot \cdot \cdot \cdot \cdot 5\cdot \cdot \cdot \cdot \cdot \cdot 6\\ & =3\text{favorable outcomes}\end{array}$$

  
  ::3 4 5 5 5 6=3 有利结果

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  For randomly choosing a date and have it land on a weekday:
  ::随意地选择日期,然后在工作日降落:

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  $$\begin{array}{rl}\text{favorable outcomes}& =\text{Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}\\ & =5\text{favorable outcomes}\end{array}$$

  
  ::星期一、星期一、星期二、星期三、星期四、星期五、星期六=5

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  To write a ratio, the favorable outcome is compared to the total outcomes. Comparing favorable outcomes to possible total outcomes is what is called theoretical probability .
  ::写一个比例,将有利结果与总结果进行比较。 将有利结果与可能的总结果进行比较就是所谓的理论概率。

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  The probability of any event is written as $\begin{array}{r}P(\text{event})\end{array}$ .
  ::任何事件的概率都以 P( 活动) 写入 。

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  For example:
  ::例如:

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  $\begin{array}{r}P(A)\end{array}$ is the probability that event $\begin{array}{r}A\end{array}$ will occur.
  ::P(A) 是事件A发生的概率。

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  $\begin{array}{r}P(\text{heads})\end{array}$ is the probability that heads will turn up on a flipped coin.
  ::P(头)是头朝硬币翻转的概率。

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  $\begin{array}{r}P(5)\end{array}$ is the probability that a number cube will turn up as 5.
  ::P(5) 是一个数字立方体为5的概率。

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  Here are two examples:
  ::以下是两个例子:

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  What is the probability of flipping heads on a coin?
  ::在硬币上翻脑袋的概率是多少?

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  $$\begin{array}{r}\text{favorable outcome}=1\text{ since there is one head on a coin}\end{array}$$

  
  ::有利的结果=1,因为硬币上有一个头

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  $$\begin{array}{r}\text{Total outcomes}=2\text{ since there is the possibility of heads or tails}\end{array}$$

  
  ::总结果=2,因为有可能出现头部或尾部

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  The answer is 1:2.
  ::答案是1:2。

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  For tossing a number cube and having it land an even number:
  ::用于抛掷一个数字立方体, 并让它降落一个偶数 :

  

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  The final answer is 1:2.
  ::最后一个答案是1:2。

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  Any time the answer can be simplified, you should do so.
  ::任何时间,只要答案可以简化,你都应该简化。

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  To find probability, follow the steps below:
  ::要发现概率,请遵循以下步骤:

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  What is the probability of the arrow landing on a yellow section?
  ::箭头降落在黄色区段的概率是多少?

  

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  First, count the number of favorable outcomes.
  ::首先,计算有利结果的数目。

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  There are 2 yellow spaces, so favorable outcomes = 2
  ::有2个黄色空格, 如此有利的结果=2

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  Next, count the number of total outcomes.
  ::其次,计算总结果的数量。

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  There are 5 spaces in all, so total outcomes = 5
  ::共有5个空间,总结果=5

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  Then, write the ratio of favorable outcomes to total outcomes.
  ::然后,写下有利结果与总结果的比例。

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  $$\begin{array}{rl}P(\text{yellow})& =\text{favorable outcomes}:\text{total outcomes}=2:5\end{array}$$

  The answer is 2:5. 
  ::P(黄)=有利结果:总结果=2:5 答案是2:5。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about B rian's worry about rain at his skate park birthday party.
  ::早些时候,你得到一个问题 关于布莱恩担心 下雨在他的滑冰公园 生日派对上。

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  What is the probability that he will escape rain at his birthday party if there have been 2 days of rain at the skate park on the same date over the past 10 years?
  ::如果在过去十年中, 同一天滑冰公园有两天的雨, 那么他在生日派对上逃雨的可能性有多大?

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  First, count the number of favorable outcomes.
  ::首先,计算有利结果的数目。

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  If there have been 2 days without rain over the past 10 years, there have been 8 days without rain which is the favorable outcome.  So favorable outcomes = 8
  ::如果在过去10年中有2天没有雨水,那么有8天没有雨水,这是有利的结果。如此有利的结果 = 8

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  Next, count the number of total outcomes.
  ::其次,计算总结果的数量。

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  There are 10 days in all, so total outcomes = 10
  ::总共有10天,所以总结果=10天

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  Then, write the ratio of favorable outcomes to total outcomes, making sure to simplify if possible. 
  ::然后,写出有利结果与总结果的比例,确保尽可能简化。

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  $$\begin{array}{rl}P(\text{no rain})& =\text{favorable outcomes}:\text{total outcomes}=8:10=4:5\end{array}$$

  The answer is 4:5.
  ::P(无雨)=有利结果:总结果=8:10=4:5 答案是4:5。

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  Example 2
  ::例2

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  What is the probability of the arrow landing on a silver or pink section when using the spinner above?
  ::当箭头在使用上面的螺旋桨时,落在银或粉色部分的概率是多少?

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  First, count the number of favorable outcomes.
  ::首先,计算有利结果的数目。

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  There are 2 silver spaces and 1 pink space, so favorable outcomes = 3
  ::有两个银色空间和一个粉色空间, 如此有利的结果 = 3

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  Next, count the number of total outcomes.
  ::其次,计算总结果的数量。

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  There are 5 spaces in all, so total outcomes = 5
  ::共有5个空间,总结果=5

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  Then, write the ratio of favorable outcomes to total outcomes
  ::然后,写下有利结果与总结果的比例

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  $$\begin{array}{r}P(\text{silver or pink})=\text{favorable outcomes}:\text{total outcomes}=3:5\end{array}$$

   The answer is 3:5.
  :粉红色或粉色)=有利结果:总结果=3:5 答案是3:5。

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  Example 3
  ::例3

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  What is the probability of rolling a 1 or a 3 on a number cube?
  ::在数字立方体上滚动 1 或 3 的概率是多少?

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  First, count the number of favorable outcomes.
  ::首先,计算有利结果的数目。

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  There is one 1 and one 3 on a number cube, so favorable outcomes = 2
  ::数字立方体上有一个1和1 3, 所以有利的结果 = 2

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  Next, count the number of total outcomes.There are 6 numbers on a number cube in all, so total outcomes = 6
  ::接下来,计算总结果的数量。 数字立方体上共有6个数字, 所以总结果= 6

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  Then, write the ratio of favorable outcomes to total outcomes, making sure to simply if possible.  
  ::然后,写下有利结果与总结果的比例,确保尽可能简单。

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  $$\begin{array}{rl}P(\text{1 or 3})& =\text{favorable outcomes}:\text{total outcomes}=2:6=1:3\end{array}$$

  
  ::P(1或3)=有利结果:总结果=2:6=1:3

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  The answer is 1:3.
  ::答案是1:3。

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  Example 4
  ::例4

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  If there are four blue marbles and one red marble in a bag, what is the probability of pulling out a red one?
  ::如果包里有四颗蓝色大理石和一颗红色大理石 那么拔出红色大理石的概率是多少?

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  First, count the number of favorable outcomes.
  ::首先,计算有利结果的数目。

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  There is one red marble in the bag, so favorable outcomes = 1
  ::袋子里有一个红大理石, 如此有利的结果=1

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  Next, count the number of total outcomes.
  ::其次,计算总结果的数量。

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  There are 5 marbles in all, so total outcomes = 5
  ::共有5个弹珠,所以总结果=5

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  Then, write the ratio of favorable outcomes to total outcomes. 
  ::然后,写下有利结果与总结果的比例。

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  $$\begin{array}{rl}P(\text{red})& =\text{favorable outcomes}:\text{total outcomes}=1:5\end{array}$$

  The answer is 1:5. 
  ::P(red)=有利结果:总结果=1:5 答案是1:5。

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  Example 5
  ::例5

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  What is the probability of pulling out a blue marble if there are four blue marbles and one red marble in a bag?
  ::如果包里有四颗蓝色大理石和一颗红色大理石,那么拔出蓝色大理石的概率是多少?

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  First, count the number of favorable outcomes.
  ::首先,计算有利结果的数目。

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  There are 4 blue marbles, so favorable outcomes = 4
  ::有4颗蓝色弹珠, 如此有利的结果=4

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  Next, count the number of total outcomes.
  ::其次,计算总结果的数量。

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  There are 5 marbles in all, so total outcomes = 5
  ::共有5个弹珠,所以总结果=5

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  Then, write the ratio of favorable outcomes to total outcomes. 
  ::然后,写下有利结果与总结果的比例。

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  $$\begin{array}{rl}P(\text{blue})& =\text{favorable outcomes}:\text{total outcomes}=4:5\end{array}$$

  The answer is 4:5. 
  ::P(蓝色)=有利结果:总结果=4:5 答案是4:5。

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  Review
  ::回顾

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  Answer each question or solve each problem as it connects to probability.
  ::回答每一个问题或解决每一个问题,因为它与概率有关。

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  For rolling a 4 on the number cube.
  ::在数字立方体上滚动一个4

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  1. How many total outcomes are there when rolling a number cube?
     ::滚动数字立方体时有多少总结果?
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  2. If the event is rolling a 4, list the favorable outcomes.
     ::如果活动正在滚动一个 4, 请列出有利的结果 。
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  3. How many favorable outcomes are there for this event?
     ::此次活动有多少有利结果?

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  For rolling a number greater than 2 on the number cube:
  ::在数字立方体上滚动超过2的数 :

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  4. If the event is rolling a 2, list the favorable outcomes.
     ::如果事件正在滚动一个 2, 请列出有利的结果 。
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  5. How many favorable outcomes are there for this event?
     ::此次活动有多少有利结果?

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  For rolling a 5 or 6 on a number cube:
  ::在数字立方体上滚动 5 或 6 :

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  6. If the event is rolling a 5 or 6, list the favorable outcomes.
     ::如果活动正在滚动 5 或 6, 请列出有利的结果 。
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  7. How many favorable outcomes are there for this event?
     ::此次活动有多少有利结果?

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  A box contains 12 slips of paper numbered 1 to 12. For randomly choosing a slip with an even number on it:
  ::框中包含12张编号为1至12的纸片,当随机选择带有偶数的纸片时:

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  8. How many total outcomes are there?
     ::有多少总结果?
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  9. If the event is picking an even number, list the favorable outcomes.
     ::如果事件正在选择一个偶数, 请列出有利的结果 。
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  10. How many favorable outcomes are there for this event?
      ::此次活动有多少有利结果?

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  A box contains 12 slips of paper numbered 1 to 12. For randomly choosing a slip with a number greater than 3:
  ::A框中包含12张编号为1至12的纸片,如果随机选择一个数字大于3的滑片:

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  11. If the event is picking a number greater than 3, list the favorable outcomes.
      ::如果事件选择的数值大于3, 请列出有利的结果 。
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  12. How many favorable outcomes are there for this event?
      ::此次活动有多少有利结果?

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  Use the following image to answer questions 16 - 25:

  

  
  ::使用以下图像回答问题 16 - 25:

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  For randomly choosing a marble and having it turn out to be orange. 
  ::因为随意选择大理石 让它变成橙色

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  13. Count the number of favorable outcomes.
      ::计算有利结果的数目。
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  14. How many total outcomes are there?
      ::有多少总结果?

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  For randomly choosing a marble and having it turn out to be large:
  ::因为随意选择大理石 并且让它变成大理石

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  15. Count the number of favorable outcomes.
      ::计算有利结果的数目。

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  For randomly choosing a marble and having it turn out to be blue:
  ::因为随意选择大理石 让它变成蓝色

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  16. Count the number of favorable outcomes.
      ::计算有利结果的数目。

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  For randomly choosing a marble and having it turn out to be small:
  ::因为随意地选择大理石 并且让大理石变得很小

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  17. Count the number of favorable outcomes.
      ::计算有利结果的数目。

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  For randomly choosing a marble and having it turn out to be orange and large:
  ::因为随机选择大理石, 并让大理石变成橙色和大:

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  18. Count the number of favorable outcomes.
      ::计算有利结果的数目。

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  Review (Answers)
  ::回顾(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 。

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  Resources
  ::资源