11.8 实验概率

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  实验概率

  

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  Jordan did a survey for his media class where he timed the length of TV advertisements (in seconds). The data he collected is shown below in the table.
  ::约旦对其媒体班进行了调查,他将电视广告的长度(以秒计)计时,收集的数据如下表所示。

  Length (s)	Frequency
  0 – 19	17
  20 – 39	38
  40 – 59	19
  60+	4

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  He needs to find the probability ratio that:
  ::他需要找到的概率比率是:

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  1. An advertisement will be between 20 and 39 seconds.
     ::广告将在20至39秒之间发布。
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  2. An advertisement will be less than 40 seconds.
     ::广告时间不到40秒

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  In this concept, you will learn to define and calculate experimental probability .
  ::在此概念中,您将学会定义和计算实验概率。

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  Experimental Probability
  ::实验概率

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  Experimental probability is probability based on doing actual experiments -flipping coins, spinning spinners, picking ping pong balls out of a jar, and so on. To compute the experimental probability of a number cube landing on 3, you would need to conduct an experiment . Suppose you were to toss the number cube 60 times.
  ::实验概率是基于实际实验的概率 — — 抛硬币、旋转旋键、从罐子中挑选乒乓球等等。 要计算数立方体在3上着陆的实验概率, 您需要进行实验。 假设您要扔60倍的立方体 。

  

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  $$\begin{array}{r}\text{Favorable outcomes}=\text{rolling a }3\end{array}$$

  
  ::有利结果=3

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  $$\begin{array}{r}\text{Total outcomes}=60\text{tosses}\end{array}$$

  
  ::结果共计=60 索谢

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  Experimental probability:
  ::实验概率 :

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  $$\begin{array}{r}\begin{array}{rcl}P(\text{event})& =& \frac{\mathrm{\#}\text{of favourable outcomes}}{\text{total}\mathrm{\#}\text{of outcomes}}\\ P(3)& =& \frac{\mathrm{\#}\text{of}{3}^{\prime }s\text{tossed}}{\text{total}\mathrm{\#}\text{of tosses}}\end{array}\end{array}$$

  
  ::P(活动)+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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  Let’s look at an example.
  ::让我们举个例子。

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  What is the experimental probability of having the number cube land on 3 when the cube is rolled 60 times?
  ::当立方体滚动60倍时, 将数字立方体降落在 3 的实验概率是多少 ?

  Trial	1	2	3	4	5	6	Total
  Raw data: 3’s	$\begin{array}{r}|\end{array}$	$\begin{array}{r}|||\end{array}$		$\begin{array}{r}|\end{array}$	$\begin{array}{r}||\end{array}$	$\begin{array}{r}||\end{array}$
  Favorable outcomes : 3’s	1	3	0	1	2	2	9
  Total Tosses							60

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  The data from the experiment shows that 3 turned up on the number cube 9 out of 60 times.
  ::实验数据显示60次中有3次出现在数字立方体9上。

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  $$\begin{array}{r}\text{Experimental Probability}=\text{favorable outcomes}:\text{total outcomes}\end{array}$$

   
  ::实验概率=有利结果:总结果

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  $$\begin{array}{r}\text{Experimental Probability}=9:60\end{array}$$

  
  ::实验概率=9:60

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  Simplified, this ratio becomes:
  ::简化后,这一比率变为:

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  $$\begin{array}{r}\text{Experimental Probability}=3:20\end{array}$$

  
  ::实验概率=3:20

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  You can see that it is only possible to calculate the experimental probability when you are actually doing experiments and counting results.
  ::您可以看到,只有在实际进行实验和计算结果时,才有可能计算实验概率。

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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 Jordan and the timed advertisements.
  ::更早之前,你被问及约旦和定时广告的问题。

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  Jordan collected his data in a graph and needs to find the probability ratio that:
  ::约旦在图表中收集了他的数据,需要找到概率比率,即:

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  1. An advertisement will be between 20 and 39 seconds.
     ::广告将在20至39秒之间发布。
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  2. An advertisement will be less than 40 seconds.
     ::广告时间不到40秒

  Length (s)	Frequency
  0 – 19	17
  20 – 39	38
  40 – 59	19
  60+	4
  Total	78

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  First, look at the data from the experiment to see how many times an advertisement was between 20 and 39 seconds. These are the favorable outcomes.
  ::首先,看看实验的数据,看看广告在20秒到39秒之间有多少次。这些都是有利的结果。

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  $$\begin{array}{r}\text{Favorable outcomes}=38\end{array}$$

  
  ::有利结果=38

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  $$\begin{array}{r}\text{Total Outcomes}=78\end{array}$$

  
  ::结果共计=78

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  Next, calculate the experimental probability.
  ::接下来,计算实验概率。

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  $$\begin{array}{r}\text{Experimental Probability}=\text{favorable outcomes}:\text{total outcomes}\end{array}$$

  
  ::实验概率=有利结果:总结果

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  $$\begin{array}{r}\text{Experimental Probability}=38:78\end{array}$$

  
  ::实验概率=38:78

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  Then, simplify the ratio.
  ::那么,简化比例。

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  $$\begin{array}{r}\text{Experimental Probability}=38:78\end{array}$$

  
  ::实验概率=38:78

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  $$\begin{array}{r}\text{Experimental Probability}=19:39\end{array}$$

  
  ::实验概率=19:39

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  The answer is 19:39.
  ::答案是19点39分

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  The experimental probability that an advertisement will be between 20 and 39 seconds is 19:39.
  ::广告在20至39秒之间的实验概率是19:39。

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  First, look at the data from the experiment to see how many times an advertisement was less than 40 seconds. These are the favorable outcomes.
  ::首先,看看实验的数据,看看广告有多少次不到40秒。这些都是有利的结果。

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  $$\begin{array}{r}\text{Favorable outcomes}=17+38=55\end{array}$$

  
  ::有利结果=17+38=55

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  $$\begin{array}{r}\text{Total Outcomes}=78\end{array}$$

  
  ::结果共计=78

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  Next, calculate the experimental probability.
  ::接下来,计算实验概率。

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  $$\begin{array}{r}\text{Experimental Probability}=\text{favorable outcomes}:\text{total outcomes}\end{array}$$

  
  ::实验概率=有利结果:总结果

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  $$\begin{array}{r}\text{Experimental Probability}=55:78\end{array}$$

  
  ::实验概率=55:78

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  The answer is 55:78.
  ::答案是55: 78

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  The experimental probability that an advertisement will be less than 40 seconds is 55:78.
  ::广告不到40秒的实验概率是55:78。

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  Example 2
  ::例2

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  Use the table to compute the experimental probability of a number cube landing on 6.
  ::使用表格计算数立方体在6上着陆的实验概率。

  Trial	1	2	3	4	5	Total
  Raw data: 6’s	$\begin{array}{r}||||\end{array}$	$\begin{array}{r}|\end{array}$	$\begin{array}{r}|\end{array}$	$\begin{array}{r}||\end{array}$	$\begin{array}{r}|\end{array}$
  Favorable outcomes: 6’s	4	1	1	2	1	9
  Total Tosses						50

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  First, look at the data from the experiment to see how many times a six turned up when rolling a number cube and also what was the total number of outcomes.  
  ::首先,看看实验中的数据 看看在滚动一个数字立方体时, 6次出现多少倍, 以及结果的总数是多少。

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  $$\begin{array}{r}\text{Favorable outcomes}=9\end{array}$$

  
  ::有利结果=9

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  $$\begin{array}{r}\text{Total Outcomes}=50\end{array}$$

  
  ::结果共计=50

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  Next, calculate the experimental probability.
  ::接下来,计算实验概率。

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  $$\begin{array}{r}\text{Experimental Probability}=\text{favorable outcomes}:\text{total outcomes}\end{array}$$

  
  ::实验概率=有利结果:总结果

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  $$\begin{array}{r}\text{Experimental Probability}=9:50\end{array}$$

  
  ::实验概率=9:50

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  The answer is 9:50.
  ::答案是9点50分

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  The experimental probability is 9:50.
  ::实验概率是9: 50

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  A number cube was tossed twenty times. The number 2 came up 3 times and the number 5 came up six times. Use this information to answer the following questions.
  ::一个数字立方体被丢弃了20次。 数字2 翻了3次, 数字5 翻了6次。 使用此信息回答下列问题 。

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  Example 3
  ::例3

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  What is the probability that the number would be a 2?
  ::数字是2的概率是多少?

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  First, look at the data from the experiment to see how many times a 2 turned up when rolling a number cube and also what was the total number of outcomes.
  ::首先,看看实验中的数据, 看看在滚动一个数字立方体时每2个出现多少次, 以及结果的总数是多少。

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  $$\begin{array}{r}\text{Favorable outcomes}=3\end{array}$$

  
  ::有利结果=3

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  $$\begin{array}{r}\text{Total Outcomes}=20\end{array}$$

  
  ::结果共计=20

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  Next, calculate the experimental probability.
  ::接下来,计算实验概率。

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  $$\begin{array}{r}\text{Experimental Probability}=\text{favorable outcomes}:\text{total outcomes}\end{array}$$

  
  ::实验概率=有利结果:总结果

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  $$\begin{array}{r}\text{Experimental Probability}=3:20\end{array}$$

  
  ::实验概率=3:20

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  The answer is 3:20.
  ::答案是3点20分

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  The experimental probability is 3:20.
  ::实验概率是3: 20

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  Example 4
  ::例4

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  What is the probability that the number would be a 5?
  ::数字是5的概率是多少?

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  First, look at the data from the experiment to see how many times a 5 turned up when rolling a number cube and also what was the total number of outcomes.
  ::首先,看看实验中的数据 看看在滚动一个数字立方体时, 5次出现多少倍, 以及结果的总数是多少。

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  $$\begin{array}{r}\text{Favorable outcomes}=6\end{array}$$

  
  ::有利结果=6

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  $$\begin{array}{r}\text{Total Outcomes}=20\end{array}$$

  
  ::结果共计=20

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  Next, calculate the experimental probability.
  ::接下来,计算实验概率。

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  $$\begin{array}{r}\text{Experimental Probability}=\text{favorable outcomes}:\text{total outcomes}\end{array}$$

  
  ::实验概率=有利结果:总结果

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  $$\begin{array}{r}\text{Experimental Probability}=6:20\end{array}$$

  
  ::实验概率=6:20

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  Then, simplify the ratio.
  ::那么,简化比例。

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  $$\begin{array}{r}\text{Experimental Probability}=6:20\end{array}$$

  
  ::实验概率=6:20

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  $$\begin{array}{r}\text{Experimental Probability}=3:10\end{array}$$

  
  ::实验概率=3:10

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  The answer is 3:10.
  ::答案是3点10分

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  The experimental probability is 3:10.
  ::实验概率是3: 10

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  Example 5
  ::例5

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  What is the probability of not rolling a 5?
  ::不滚动5的概率是多少?

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  First, look at the data from the experiment to see how many times a 5 did not turn up when rolling a number cube and also what was the total number of outcomes. You know that it 5 did turn up 6 times in the 20 tosses.
  ::首先,看看实验中的数据,看看在滚动一个数字立方体时,每5个数据有多少次没有出现, 以及结果总数是多少。你知道,每20个倾斜点中,5个数据确实出现6次。

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  $$\begin{array}{r}\text{Favorable outcomes}=20-6=14\end{array}$$

  
  ::有利结果=20-6=14

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  $$\begin{array}{r}\text{Total Outcomes}=20\end{array}$$

  
  ::结果共计=20

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  Next, calculate the experimental probability.
  ::接下来,计算实验概率。

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  $$\begin{array}{r}\text{Experimental Probability}=\text{favorable outcomes}:\text{total outcomes}\end{array}$$

  
  ::实验概率=有利结果:总结果

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  $$\begin{array}{r}\text{Experimental Probability}=14:20\end{array}$$

  
  ::实验概率=14:20

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  Then, simplify the ratio.
  ::那么,简化比例。

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  $$\begin{array}{r}\text{Experimental Probability}=14:20\end{array}$$

  
  ::实验概率=14:20

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  $$\begin{array}{r}\text{Experimental Probability}=7:10\end{array}$$

  
  ::实验概率=7:10

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  The answer is 7:10.
  ::答案是7点10分

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  The experimental probability is 7:10.
  ::实验概率是7: 10

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  Review
  ::回顾

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  Find the probability for rolling less than 4 on the number cube.
  ::在数字立方体上找到滚动小于 4 的概率 。

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  1. List each favorable outcome.
  ::1. 列出每项有利结果。

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

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  3. Count the number of total outcomes.
  ::3. 计算总结果的数目。

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  4. Write the probability.
  ::4. 记录概率。

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  Find the probability for rolling 1 or 6 on the number cube.
  ::在数字立方体上找到滚动 1 或 6 的概率 。

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  5. List each favorable outcome.
  ::5. 列出每一有利结果。

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  6. Count the number of total outcomes.
  ::6. 计算总结果的数目。

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  7. Write the total number of outcomes.
  ::7. 记录结果总数。

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  8. Write the probability.
  ::8. 记录概率。

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  A box contains 12 slips of paper numbered 1 to 12. Find the probability for randomly choosing a slip with a number less than 4 on it.
  ::框中包含12张编号为1至12的纸片,找到随机选择一个号码小于4的滑纸的概率。

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  9. List each favorable outcome.
  ::9. 列出每一有利结果。

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

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  11. Count the number of total outcomes.
  ::11. 计算总结果的数目。

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  12. Write the probability.
  ::12. 记录概率。

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  Use the table to answer the questions. Express all ratios in simplest form.
  ::使用表格回答问题。以最简单的形式表示所有比率。

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  Use the table to compute the experimental probability of flipping a coin and having it land on heads.
  ::使用该表来计算翻硬币的实验概率, 并让硬币在头顶上降落 。

  Trial	1	2	3	4	5	6	Total
  Raw data(heads)	$\begin{array}{r}\bcancel{||||}\end{array}$	$\begin{array}{r}\bcancel{||||}|\end{array}$	$\begin{array}{r}\bcancel{||||}|\end{array}$	$\begin{array}{r}|||\end{array}$	$\begin{array}{r}\bcancel{||||}|\end{array}$	$\begin{array}{r}\bcancel{||||}\end{array}$
  Number of heads	5	6	6	3	6	5	31
  Total number of flips	10	10	10	10	10	10	60

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  13. How many favorable outcomes were there in the experiment?
  ::13. 实验中有多少有利结果?

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  14. How many total outcomes were there in the experiment?
  ::14. 试验中有多少总结果?

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  15. What was the experimental probability of the coin landing on heads?
  ::15. 硬币在头顶上着陆的实验概率是多少?

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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 。