7.6 宗学实验

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  分子实验

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  Binomial experiments are very popular for studies because the probability of one possibility or the other can be calculated quickly and accurately. How do you identify a binomial experiment? Can an experiment that is not binomial be easily converted into a binomial experiment?
  ::二进制实验在研究中非常受欢迎,因为一种可能性的概率可以快速和准确地计算。您如何识别二进制实验?非二进制实验可以很容易地转换成二进制实验吗?

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  Look to the end of the lesson for the answer.
  ::寻找教训的结尾 以找到答案。

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  Binomial Experiments 
  ::分子实验

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  Binomial experiments give rise to binomial random variables , which will be the topic of our next couple of lessons. A binomial experiment is a very specific type of experiment. In order to be a binomial experiment, there are four qualifications that the experiment must meet:
  ::二进制实验产生二进制随机变数,这是我们接下来几个课程的主题。二进制实验是一种非常特殊的实验类型。要成为二进制实验,实验必须满足四个条件:

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  1. There must be a fixed number of trials . The experiment cannot just be to roll a die until you get a 2, because the number of rolls (trials) is not fixed.
     ::试验必须有一个固定的试验次数。 实验不能仅仅在获得 2 之前滚死, 因为卷数( 审判) 没有固定 。
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  2. Each trial must be independent of the others. You cannot have a situation like “If you flip a coin and get heads, flip twice more, and if you get tails, flip three more times”.
     ::每个审判必须独立于其他审判。 你不能有这样的情况:“如果你翻硬币并获得头部,翻两次,如果你得到尾巴,翻三次”。
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  3. Each trial must have a “success” and a “failure”. Depending on the trial, these may be identified as “yes” and “no” or “0” and “1” or “black” and “white”, etc. However, from a statistics standpoint, the outcome you are studying is generally called the “success” and the other is called “failure”.
     ::每个审判必须有一个“成功”和“失败”,视审判情况而定,这些可以被确定为“是”和“否”或“0”和“1”或“黑人”和“白人”等,然而,从统计的角度来看,你正在研究的结果一般称为“成功”,另一个称为“失败”。
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  4. The probability of success must be the same for all trials. The experiment cannot be 10 trials of pulling and keeping a card from a deck to see how many are hearts, because the probability of getting a heart would change each trial. To make this a binomial experiment, you need to replace the card each time.
     ::成功概率必须是所有试验的相同概率。 实验不能是从甲板上拉动和保留一张牌以查看有多少是心的10次试验, 因为获得心脏的概率会改变每一次试验。 要将它变成二进制实验, 您需要每次更换牌。

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  Identifying Binomial Random Variables 
  ::识别分义随机变量

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  1. If a fair coin if flipped 10 times, and  $\begin{align*}T\end{align*}$ is the number of tails, is $\begin{align*}T\end{align*}$ a binomial random variable ?
  ::1. 如果一个公平硬币翻了10次,而T是尾数,那么T是一个二进制随机变数吗?

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  Yes,  $\begin{align*}T\end{align*}$ is a binomial random variable, and this is a binomial experiment. It meets all four qualifications:
  ::是的,T是一个二进制随机变量,这是一个二进制实验。它符合所有四种条件:

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  1. There is a specific number of trials: 10 flips
     ::有具体数目的审判:10次翻案
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  2. Trials are independent: the outcome of one coin flip does not affect the next flip
     ::审判是独立的:一个硬币翻转的结果不影响下一个翻转
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  3. There are only two possible outcomes: a “success” and “failure”. Since we are counting tails, every tails is a “success” and every heads is a “failure”
     ::可能的结果只有两种:“成功”和“失败”,因为我们在计算尾巴,每个尾巴都是“成功”,每个头都是“失败”
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  4. The probability is the same for all trials: The probability of getting tails is always 50% if flipping a fair coin
     ::所有试验的概率都是一样的:如果抛出一个公平的硬币,获得尾巴的概率总是50%。

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  2. If Trina designates  $\begin{align*}Y\end{align*}$ to be the number of yellow marbles she gets during nine trials of randomly pulling 1 marble from a bag filled with marbles of various colors and returning it, is  $\begin{align*}Y\end{align*}$ a random variable ? Is it binomial?
  ::2. 如果Trina指定Y为她9次随机抽取1大理石的黄色大理石,从装满各种颜色的大理石的袋子中抽取1大理石,然后还回去,Y是一个随机变量吗?它是二元论的吗?

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  Yes, $\begin{align*}Y\end{align*}$ is a random variable, since it is the random numerical result of a limited number of independent trials of an experiment. It is also binomial, since each of the limited trials is independent, has a success/failure (yellow/not yellow), and has the same probability of success.
  ::是的,Y是一个随机变量,因为它是一个实验独立试验数量有限的随机数字结果。它也是二元论,因为每个有限的试验都是独立的,成功/失败(黄/黄/黄),成功概率相同。

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  Finding Unknown Values 
  ::查找未知值

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  If  $\begin{align*}N\end{align*}$ is the number of nines you get when rolling two standard dice three times:
  ::如果 N 是滚动双标准骰子三次时得到的 9 个数 :

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  1. Is $\begin{align*}N\end{align*}$ a binomial random variable?
     ::一个二进制随机变量吗?
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  2. What are the possible values of $\begin{align*}N\end{align*}$ ?
     ::N的可能价值是什么?
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  3. Create a histogram or pie chart showing the probability distribution of $\begin{align*}N\end{align*}$ .
     ::创建直方图或饼图,显示 N 的概率分布 。

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  a. Is  $\begin{align*}N\end{align*}$  a binomial random variable?
  ::a. N是一个二进制随机变数吗?

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  $\begin{align*}N\end{align*}$ is a binomial random variable, because it is the result of a specific and limited number of independent trials of a random process and each outcome is either nine or not nine.
  ::N是一个二进制随机变量,因为它是随机过程的个别和有限的独立试验的结果,每个结果有9个或9个。

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  b. What are the possible values of  $\begin{align*}N\end{align*}$ ?
  ::b. N的可能值是什么?

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  Since you could only roll a total of 9 once each trial,  $\begin{align*}N\end{align*}$ could be 0, 1, 2, or 3.
  ::因为每次审判,你只能卷9次, N可能是0, 1, 2, 或3次。

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  c. Create a histogram or pie chart showing the probability distribution of  $\begin{align*}N\end{align*}$ . 
  ::c. 创建直方图或饼图,显示N的概率分布。

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  The probabilities of each of the possible values of $\begin{align*}N\end{align*}$ would be:
  ::每一可能值的概率为:

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  (see the lesson: Understanding Discrete Random Variables, Example C, for the calculations)
  :见教训:了解分辨随机变量,用于计算)

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  • $\begin{align*}N=0: \frac{512}{729}\end{align*}$
    ::N=0:512729
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  • $\begin{align*}N=1: \frac{192}{729}\end{align*}$
    ::N=1:192729
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  • $\begin{align*}N=2: \frac{24}{729}\end{align*}$
    ::N=2:24729
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  • $\begin{align*}N=3: \frac{1}{729}\end{align*}$
    ::N=3:1729

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  A pie chart would look like this: (note that total probability = $\begin{align*}\frac{512}{729} + \frac{192}{729} + \frac{24}{729} + \frac{1} {729} = \frac{729}{729}\end{align*}$ ).
  ::馅饼图表将看起来是这样的注意总概率=512729+192729+24729+1729=729729)。

  

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  Earlier Problem Revisited
  ::重审先前的问题

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  Binomial experiments are very popular for studies because the probability of one possibility or the other can be calculated quickly and accurately. How do you identify a binomial experiment? Can an experiment that is not binomial be easily converted into a binomial experiment?
  ::二进制实验在研究中非常受欢迎,因为一种可能性的概率可以快速和准确地计算。您如何识别二进制实验?非二进制实验可以很容易地转换成二进制实验吗?

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  A binomial experiment must consist of a limited number of independent trials, where each trial outcome is either a success or a failure, and each trial has the same probability of success as all other trials.
  ::二元实验必须包括数量有限的独立审判,其中每个审判结果要么成功要么失败,每个审判成功的可能性与所有其他审判相同。

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  A non-binomial experiment can often be viewed as binomial by carefully stating the outcome of each trial in a binomial format. For example, a non-binomial experiment might be “Count the number of heads and tails resulting from 8 flips of a fair coin”. Viewed as a binomial experiment, the same results could be collected from “How many tails do you get by flipping a fair coin 8 times”? You could then subtract the result from 8 to get the number of “not tails”, e.g. “heads”.
  ::非二元实验通常可以被视为二元实验,以二元形式仔细说明每次试验的结果,例如,非二元实验可以是“计算公平硬币8个翻转产生的头部和尾部的数量”。 作为一种二元实验,可以从“翻开公平硬币8次,有多少尾部会得到多少次”中收集同样的结果。 然后,你可以从8个中减去结果,以获得“非尾部”的数目,例如“头部”。

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  Examples 
  ::实例

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  Example 1
  ::例1

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  Mariska spins a spinner 40 times, recording the number of 4’s she gets. Is this a binomial experiment?
  ::Mariska旋转了一个旋转器40倍,记录了她得到的四分位数。这是二进制实验吗?

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  Yes, this is a binomial experiment because Mariska is conducting a limited number of independent random "4" or "not 4" trials, and the probability of spinnig a "4" does not change.
  ::是的,这是一个二进制实验, 因为Mariska正在进行数量有限的 独立的随机“4”或“非4”试验, 脊柱“4”的概率没有改变。

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  Example 2
  ::例2

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  Heidi has a bag containing 4 blue, 3 green, 5 red, and 7 yellow marbles. She defines a trial as pulling a marble, recording the color, and replacing it. She records the number of trials it takes to pull a green marble. Is this a binomial experiment?
  ::海蒂有一个装有4个蓝色、3个绿色、5个红色和7个黄色大理石的袋子。 她将试验定义为拉大理石、记录颜色并替换它。 她记录了拉绿色大理石需要多少试验。 这是二元论实验吗?

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  No, Heidi is not conducting a binomial experiment because the number of trials is not specified, she just keeps pulling until she gets a green.
  ::不,海蒂没有进行二进制实验 因为试验数量没有说明 她只是不停地拉 直到她得到绿色。

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  Example 3
  ::例3

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  Evan notes that 24% of online game players he polled are between 30 and 39 years old. Evan decides to create a team of players from that age range by randomly choosing names from among those he polled, keeping each one he chooses that is in his/her 30’s. If he chooses a name only 10 times, no matter the number of players he gets, is this a binomial experiment?
  ::埃文指出,他所选的网上游戏玩家中,24%的人年龄在30至39岁之间。 埃文决定通过随机从他所选的球员中选取名字,将他所选的每一个名字保留在30年代。 如果他只选了10次名字,不管他得到多少球员,这是二元论实验吗?

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  No, Evan is not conducting a binomial experiment because the probability that a random player will be between 30 and 39 changes each tim ehe keeps one for his team. 
  ::不,埃文不是在做一个二进制实验 因为随机玩家在30到39之间 的概率会改变 每一个timehe 为其团队保留一个。

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  Review 
  ::回顾

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  For questions 1-12, state that a particular experiment is or why it is not binomial:
  ::对于问题1-12, 说明某一特定实验是二元实验, 或为什么它不是二元实验:

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  1. A spinner has a 35% probability of landing on blue. Let  $\begin{align*}B\end{align*}$ be the number of blues spun in 5 spins.
     ::旋转器在蓝色上着陆的概率为35%。 B 是蓝色在5个旋转时的旋转次数 。
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  2. A bag contains 6 blue, 4 green, and 3 red candies. Let  $\begin{align*}G\end{align*}$ be the number of green candies you pull out and eat in 5 trials.
     ::包里装有6个蓝色、4个绿色和3个红色糖果。让G成为5次试验中你拿出来吃绿色糖果的数量。
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  3. One trial of an experiment consists of pulling a random card from a standard deck, noting it, and replacing it, you conduct 12 trials.
     ::实验的试验之一是从标准甲板上抽取一张随机卡片,注明并替换它,进行12次试验。
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  4. One trial consists of pulling two cards from a standard deck, noting them, and replacing them. Let  $\begin{align*}T\end{align*}$ be the number of trials until you pull two face cards at the same time.
     ::其中一项审判包括从标准甲板上拉出两张卡片,注明并替换它们。让T成为审判的数量,直到你同时拉出两张脸卡。
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  5. A 20-sided die is rolled ten times, and  $\begin{align*}S\end{align*}$ is the number of sevens rolled.
     ::20方死亡是十倍,七方死亡是七方死亡。
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  6. Assume that 15% of word game players create at least 12 words out of 50 that have more than 5 letters, and you let  $\begin{align*}W\end{align*}$ be the number of letters in words from 20 trials of 1 game each.
     ::假设15%的单词游戏玩家在50个有超过5个字母的单词中 创造出至少12个字, 你让W成为20个字数的字母数, 每个字数为1个。
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  7. A die is rolled 20 times. What is the probability of rolling a 1 exactly 5 times?
     ::死亡是滚动20次。滚动1次的概率是多少? 确切的说是5次?
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  8. You plan on choosing students (with replacement) from a population of 28, 17 of which are Juniors. You want to know how many will have to be picked before getting a Junior.
     ::您计划从28人中选择学生( 替换学生 ) , 其中17人是青年。 您想知道在获得青年之前需要挑选多少人 。
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  9. A new reality show is so popular that an estimated 47% of households watch it every week. You choose 20 households at random. Let  $\begin{align*}X\end{align*}$ be the number of households watching the show.
     ::新的现实节目非常流行,估计有47%的家庭每周都看。您随机选择20个家庭。让X成为观看节目的家庭数量。
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  10. $\begin{align*}H\end{align*}$  is the number of heads tallied over ten flips of a fair coin.
      ::H 是一个公平硬币的十张硬币的 头数。
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  11. $\begin{align*}F\end{align*}$  is the number of 5’s you roll before rolling a 6, on a standard die.
      ::F是按标准死亡时在滚动6之前滚动的5个数字。
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  12. $\begin{align*}O\end{align*}$  is the number of 1’s you roll in fifteen rolls of a standard die.
      ::O是标准死亡的15卷中你滚动的1号数。

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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 。