11.10 查明重叠、脱节和补充活动

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  识别重叠、脱节和补充活动

  

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  Instead of staying home to watch cartoons on Saturday morning, Stella decided to go with her dad to his squash club. However, when she got there she found there wasn’t much to do but look at all the different squash players. So Stella decided to make a game of it. Stella has black hair and blue eyes, so she decided to count how many of the people she saw also had those characteristics. By the end of her dad’s game, Stella had counted 27 members in total. Of those 27, 19 had black hair, 14 had blue eyes, and 11 had both black hair and blue eyes. How can Stella tell if these events are disjoint or overlapping? 
  ::斯特拉没有在周六早上呆在家里看漫画,而是决定和她父亲去他的壁球俱乐部。 但是,当她到达那里时,她发现除了看所有不同的壁球运动员之外没有什么事情要做。 因此斯特拉决定玩一个游戏。 斯特拉有黑头发和蓝眼睛,因此她决定计算她看到的人中有多少人也有这些特点。 在她父亲的游戏结束时,斯特拉总共统计了27名成员。 在这27人中,19人有黑头发,14人有蓝眼睛,11人有黑头发和蓝眼睛。 斯特拉如何知道这些事件是否脱节或重叠?

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  In this concept, you will learn to identify and distinguish between overlapping, disjoint and complementary events .
  ::在这一概念中,你们将学会识别和区分重叠、脱节和互补事件。

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  Disjoint and Complementary Events
  ::中断和补充性活动

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  When you spin a spinner or roll a die to calculate a probability , some probabilities have events in common and some don’t. This is where you can begin to talk about identifying disjoint events . Disjoint events are events that don’t have any outcomes in common.
  ::当你旋转一个旋转器或滚动一个死亡来计算概率时,有些概率是共同的,有些则不是。 你可以在这里开始谈论识别脱节事件。 脱节事件是没有任何共同结果的事件。

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  Take a look at this situation.
  ::看看这个情况。

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  Consider spinning this spinner:
  ::考虑旋转此旋转器 :

  

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  • Event $\begin{align*}A\end{align*}$ : {yellow} 
    ::事件A:{黄}
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  • Event $\begin{align*}B\end{align*}$ : {blue}
    ::事件B:{蓝}

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  Events $\begin{align*}A\end{align*}$  and $\begin{align*}B\end{align*}$  are disjoint events because they have no outcomes in common - the arrow either lands on blue or yellow.
  ::事件A和B是相互脱节的事件,因为它们没有共同的结果----箭头降落在蓝色或黄色上。

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  You can use a to show when events overlap and when they don’t overlap. A Venn diagram shows a way of representing whether events have any outcomes in common. It has round shapes that overlap or don’t overlap. The Venn diagram for disjoint events shows no overlap between the two events.
  ::您可以使用 a 来显示事件重叠的时间和事件不重叠的时间。 Venn 图表显示了事件是否有共同结果的表达方式。 它的圆形会重叠或不重叠。 用于脱节事件的 Venn 图表显示两个事件之间没有重叠。

  

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  Not all events are disjoint. There are many events that are connected to each other.
  ::并非所有事件都是脱节的,有许多事件是相互联系的。

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  Let’s look at another example.
  ::让我们再看看另一个例子。

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  Consider this spinner and the events   $\begin{align*}R\end{align*}$  (red) and $\begin{align*}T\end{align*}$  (top).
  ::想想这个旋转符和事件 R( red) 和 T( top) 。

  

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  • Event $\begin{align*}R\end{align*}$ : { red-top , red-bottom }
    ::事件 R: {红顶, 红底 }
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  • Event $\begin{align*}T\end{align*}$ : { red-top , blue-top }
    ::事件 T: {红顶, 蓝顶}

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  Clearly, both events share an outcome −  red-top −  so the two are called overlapping events . The Venn diagram for overlapping events shows that the two events overlap, or share 1 or more outcomes.
  ::显然,这两个事件都有一个共同的结果 — — 红色顶端 — — 因此这两个事件被称为重叠事件。 重叠事件的文恩图表显示,这两个事件重叠,或者分享一个或一个以上的结果。

  

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  Complementary events are events whose probability sum adds up to 1.0 or 100 percent.
  ::补充性事件是概率总和达1.0%或100%的事件。

  

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  Events $\begin{align*}Y\end{align*}$  and $\begin{align*}G\end{align*}$  in this spinner above are complementary.
  ::上面的“Y”和“G”事件是相辅相成的。

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  $$\begin{align*}P(\text{yellow})+P(\text{green})=1\end{align*}$$
  ::P(黄)+P(绿色)=1

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  Complementary events are either-or events. Either the spinner above lands on green or it lands on yellow. There are no other outcomes.
  ::补充性活动要么是活动,要么是绿地上的旋翼者,要么是黄地上。没有其他结果。

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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  bored Stella at the squash club.
  ::早些时候,你在壁球俱乐部 遇到一个无聊的史黛拉的问题

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  Stella needs to determine if the events of black hair and blue eyes are disjoint or overlapping.
  ::Stella需要确定黑头发和蓝眼睛的事件是否脱节或重叠。

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  First, let’s list all of the numbers in Events $\begin{align*}A\end{align*}$  and $\begin{align*}B\end{align*}$ .
  ::首先,让我们列出事件A和B的所有数字。

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  Event $\begin{align*}A\end{align*}$ : black hair
  ::事件A:黑头发

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  Event $\begin{align*}B\end{align*}$ : 4, blue eyes
  ::事件B:4,蓝眼4,蓝色眼睛

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  Next, remember the definitions of disjoint and overlapping events.
  ::接下来,请记住对脱节和重叠事件的定义。

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  Disjoint events are events that don’t have any outcomes in common.
  ::与外界隔绝的事件是没有任何共同结果的事件。

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  Overlapping events are events that have outcomes in common.
  ::重叠事件是具有共同结果的事件。

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  Then, determine whether the events are disjoint or overlapping. You can do this by drawing a Venn diagram that shows if any events are common to the two sets.
  ::然后, 确定事件是否脱节或重叠。 您可以绘制 Venn 图表, 显示两组事件是否相同 。

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  Remember that there are 19 people with black hair and 11 also have blue eyes, therefore 8 have eyes of a different color.
  ::记得有19个黑头发人,11个蓝眼睛人, 因此8个眼睛的颜色不同。

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  Also, 14 people have blue eyes and 11 of these also have black hair, therefore 3 have hair of a different color.
  ::另外,14人有蓝眼睛,其中11人也有黑头发,因此3人有不同颜色的头发。

  

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  Looking at the Venn diagram above, it is clear that these events are overlapping as they have outcomes in common.
  ::从上面的文恩图看,很明显,这些事件是重叠的,因为它们有共同的结果。

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  Example 2
  ::例2

  

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  Are event red and event blue complementary or disjoint events? Why?
  ::红色事件和蓝色事件是补充性事件还是脱节事件? 为什么?

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  First, remember the definitions of overlapping and complementary events.
  ::首先,记住重叠和互补活动的定义。

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  Overlapping events are events that have outcomes in common.
  ::重叠事件是具有共同结果的事件。

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  Complementary events are events whose probability sum adds up to 1 or 100%.
  ::补充事件是概率加起来等于1%或100%的事件。

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  Next, look at the spinner. 50% of the spinner is red and 50% is blue. There is no section on the spinner that has both red and blue.
  ::下一个,看看旋转器。 50%的旋转器是红色的, 50%是蓝色的。 旋转器上没有有红色和蓝色的部分 。

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  The answer is complementary. Therefore, the two events are complementary.
  ::答案是相辅相成的,因此,这两个事件是相辅相成的。

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  Example 3
  ::例3

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  For the spinner below, if event $\begin{align*}A\end{align*}$  is bottom and event $\begin{align*}B\end{align*}$  is blue, are the events disjoint or overlapping. Are the events complementary?
  ::对于下面的旋转符来说,如果事件A是底部,事件B是蓝色,事件是脱节或重叠。事件是互补的吗?

  

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  First, remember the definitions of disjoint and overlapping events.
  ::首先,记住脱节和重叠事件的定义。

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  Disjoint events are events that don’t have any outcomes in common.
  ::与外界隔绝的事件是没有任何共同结果的事件。

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  Overlapping events are events that have outcomes in common.
  ::重叠事件是具有共同结果的事件。

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  Complementary events are events whose probability sum adds up to 1 or 100%.
  ::补充事件是概率加起来等于1%或100%的事件。

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  Next, look at the spinner to determine what type of event is occurring.
  ::接下来,看看螺旋体以确定正在发生何种类型的事件。

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  Event $\begin{align*}A\end{align*}$  is the sections with the word bottom.
  ::事件A是带有单词底部的章节 。

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  Event $\begin{align*}B\end{align*}$  includes the blue sections.
  ::B事件包括蓝色部分。

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  Then determine whether these events are complementary, disjoint or overlapping.
  ::然后确定这些事件是相辅相成、脱节还是重叠。

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  They are not complementary because the red - top is not included in the two events.
  ::它们不是相辅相成的,因为红色 - 顶部不包括在这两个事件之中。

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  They are not disjoint as there is a blue - bottom.
  ::它们不是脱节的,因为有一个蓝色的底部。

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  Therefore the events are overlapping.
  ::因此,事件是重叠的。

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  Example 4
  ::例4

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  If event $\begin{align*}A\end{align*}$  includes the numbers 1, 6, 7, and 9 and event $\begin{align*}B\end{align*}$  includes the numbers 2, 3, 4, 5, and 8. Is this an example of disjoint or overlapping events?
  ::如果事件A包括编号1、6、7和9,事件B包括编号2、3、4、5和8,这是脱节或重叠事件的例子吗?

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  First, let’s list all of the numbers in Events $\begin{align*}A\end{align*}$  and $\begin{align*}B\end{align*}$ .
  ::首先,让我们列出事件A和B的所有数字。

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  Event $\begin{align*}A\end{align*}$ : 1, 6, 7, 9.
  ::事件A: 1,6,7,9。

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  Event $\begin{align*}B\end{align*}$ : 2, 3, 4, 5, 8.
  ::事件B: 2, 3, 4, 5, 8。

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  Next, remember the definitions of disjoint and overlapping events.
  ::接下来,请记住对脱节和重叠事件的定义。

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  Disjoint events are events that don’t have any outcomes in common.
  ::与外界隔绝的事件是没有任何共同结果的事件。

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  Overlapping events are events that have outcomes in common.
  ::重叠事件是具有共同结果的事件。

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  Then, determine whether the events are disjoint or overlapping. You can do this by drawing a Venn diagram that shows if any events are common to the two sets.
  ::然后, 确定事件是否脱节或重叠。 您可以绘制 Venn 图表, 显示两组事件是否相同 。

  

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  The answer is that these events are disjoint as there are no numbers common to both events.
  ::答案是,这些事件是脱节的,因为这两个事件没有共同的数字。

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  Example 5
  ::例5

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  If Event $\begin{align*}A\end{align*}$  is the set of even numbers from 0 to 20 and Event $\begin{align*}B\end{align*}$  is the set of numbers that are multiples of 4 and less than 30, Is this an example of disjoint or overlapping events?
  ::如果事件A是0至20的偶数组,而事件B是4和30以下的倍数组,这是脱节或重叠事件的例子吗?

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  First, let’s list all of the numbers in Events $\begin{align*}A\end{align*}$  and $\begin{align*}B\end{align*}$ .
  ::首先,让我们列出事件A和B的所有数字。

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  Event $\begin{align*}A\end{align*}$ : 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.
  ::事件A: 2,4,6,8,10,12,14,16,18,20。

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  Event $\begin{align*}B\end{align*}$ : 4, 8, 12, 16, 20, 24, 28.
  ::事件B:4、8、12、16、20、24、28。

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  Next, remember the definitions of disjoint and overlapping events.
  ::接下来,请记住对脱节和重叠事件的定义。

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  Disjoint events are events that don’t have any outcomes in common.
  ::与外界隔绝的事件是没有任何共同结果的事件。

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  Overlapping events are events that have outcomes in common.
  ::重叠事件是具有共同结果的事件。

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  Then, determine whether the events are disjoint or overlapping. You can do this by drawing a Venn diagram that shows if any events are common to the two sets.
  ::然后, 确定事件是否脱节或重叠。 您可以绘制 Venn 图表, 显示两组事件是否相同 。

  

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  The answer is that these events are overlapping as there are numbers common to both events.
  ::答案是,这些事件是重叠的,因为这两个事件都有共同的数字。

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  Review
  ::回顾

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  Solve the problems. For overlapping events, tell which events overlap.
  ::解决问题。对于重叠事件,请说明哪些事件重叠。

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  1. For a flip of a coin, are events $\begin{align*}H\end{align*}$  (heads) and $\begin{align*}T\end{align*}$  (tails) disjoint or overlapping?
  ::1. 就一枚硬币而言,事件H(头)和T(尾)是否脱节或重叠?

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  2. For a flip of a coin, are events $\begin{align*}H\end{align*}$  (heads) and $\begin{align*}T\end{align*}$  (tails)  complementary or non-complementary?
  ::2. 就一枚硬币而言,事件H(头)和T(尾)是相辅相成还是不互补?

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  3. Why?
  ::3. 为什么?

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  4. For a single toss of a number cube, are $\begin{align*}E\end{align*}$  (even) and $\begin{align*}T\end{align*}$  (3) disjoint events or overlapping events?
  ::4. 对于一个数字立方体的一整块,E(偶数)和T(3)是否是脱节事件或重叠事件?

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  5. Why?
  ::5. 为什么?

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  6. For a single toss of a number cube, are $\begin{align*}E\end{align*}$  (even) and $\begin{align*}S\end{align*}$  (6) disjoint events or overlapping events?
  ::6. 对于一个数字立方体的一整块,E(偶数)和S(6)是脱节事件还是重叠事件?

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  7. For a single toss of a number cube, are $\begin{align*}G3\end{align*}$  (greater than 3) and $\begin{align*}O\end{align*}$  (odd) disjoint events or overlapping events?
  ::7. 对于一个数字立方体的一振,G3(大于3)和O(od)是脱节事件还是重叠事件?

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  8. For a single toss of a number cube, are $\begin{align*}E\end{align*}$  (even) and $\begin{align*}O\end{align*}$  (odd) complementary events or non-complementary events?
  ::8. 对于一个数字立方体的一小块,E(偶数)和O(偶数)是补充性活动还是非补充性活动?

  

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  9. For a single spin, are $\begin{align*}B\end{align*}$  (blue) and $\begin{align*}G\end{align*}$  (green) disjoint events or overlapping events?
  ::9. 就一盘子而言,B(蓝色)和G(绿色)事件是脱节还是重叠事件?

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  10. For a single spin, are $\begin{align*}G\end{align*}$  (green) and $\begin{align*}L\end{align*}$  (left) disjoint events or overlapping events?
  ::10. 就单一旋转而言,G(绿色)和L(左)是脱节事件还是重叠事件?

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  11. For a single spin, are $\begin{align*}Y\end{align*}$  (yellow) and $\begin{align*}R\end{align*}$  (red) complementary or non-complementary events?
  ::11. 就单一旋转而言,Y(黄色)和R(红色)是互补还是非补充性事件?

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  12. For a single spin, are $\begin{align*}R\end{align*}$  (right) and $\begin{align*}L\end{align*}$  (left) complementary or non-complementary events?
  ::12. 就单一旋转而言,R(右)和L(左)是互补还是非互补事件?

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  13. For a light switch, are ON and OFF disjoint or overlapping events?
  ::13. 对于灯开关而言,是否在非联邦部队内或境外部队内发生脱节或重叠事件?

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  14. For a light switch, are ON and OFF complementary or non-complementary events?
  ::14. 对于灯开关而言, " 在线 " 和 " 外部论坛 " 是补充性还是非补充性活动?

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  15. For an oven, are ON and OFF disjoint or overlapping events?
  ::15. 就烤箱而言,在海外或海外办事处发生脱节或重叠事件吗?

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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 。