12.4 相互排斥的活动

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  相互排斥活动

  

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  Elizabeth will be the lead dancer in a ballet performance tonight.  She practiced all afternoon and successfully accomplished 25 pirouettes (spins) out of 30 attempts.  She feels as though if she walks around the block twice before she leaves for the performance, she will successfully complete her pirouettes during the performance.  Are the two events, pirouettes and walking around the block, disjoint events ?  In other words, are they mutually exclusive ?
  ::Elizabeth将是今晚芭蕾舞表演的主舞者。她练习了整个下午,在30次尝试中成功完成了25次。她觉得如果她离开表演前在街区走过两次,她将在表演期间成功完成她的芭蕾舞表演。这两个活动,即双轮舞和在街上走过,是脱节吗?换句话说,它们是相互排斥的吗?

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  In this concept, you will learn about probability and mutually exclusive events.
  ::在这个概念中,你将了解概率和相互排斥的事件。

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  Mutually Exclusive Events
  ::相互排斥活动

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  Disjoint events are two events that have no outcomes in common. They are mutually exclusive of each other.  For example, for a pregnant woman, here are two events that have no outcomes in common.
  ::脱节事件是两个没有共同结果的事件,它们相互排斥,例如,对孕妇来说,这里是两个没有共同结果的事件。

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  • Event B : Having a baby boy.
    ::事件B:生了一个小男孩。
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  • Event G : Having a baby girl.
    ::事件G:生一个女婴。

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  Disjoint events are either-or events.  For example, consider flipping a coin. The two events–flipping heads or flipping tails–have no outcomes in common. You either flip heads or you flip tails.
  ::脱节事件要么是事件,要么是事件。例如,考虑翻硬币。 这两项事件 — — 翻转头部或翻转尾巴 — — 没有共同的结果。 你要么翻转头部,要么翻转尾巴。

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  The probability of one of the two disjoint events occurring is just the sum of the probabilities of the events. Since the probability of flipping heads is $\begin{align*}\frac{1}{2}\end{align*}$ and the probability of flipping tails is $\begin{align*}\frac{1}{2}\end{align*}$ :
  ::发生两起脱节事件的概率之一只是事件的概率之和。 因为翻转头的概率是12,翻转尾的概率是12:

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  $$\begin{align*}P (\text{heads or tails}) &= P (\text{heads}) + P (\text{tails})\\ &= \frac{1}{2} + \frac{1}{2}\\ &= 1\end{align*}$$
  ::P(头或尾)=P(头)+P(尾)=12+12=1

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  In other words, all possibilities are covered. The probability of either heads or tails is 1–it will be either heads or tails 100 percent of the time.
  ::换句话说,所有可能性都被覆盖了。 正面或尾巴的概率是1 — — 将是正面或尾巴的100%。

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

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  For a single spin, are events $\begin{align*}R (\text{red})\end{align*}$ and $\begin{align*}B (\text{blue})\end{align*}$ disjoint events?
  ::R(red)和B(B(blue)事件是否因单一旋转而脱节?

  

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  First, to see if $\begin{align*}P (\text{red})\end{align*}$ and $\begin{align*}P (\text{blue})\end{align*}$  are disjoint events, make a list of the outcomes of $\begin{align*}P (\text{red})\end{align*}$ and $\begin{align*}P (\text{blue})\end{align*}$ :
  ::首先,为了看P(red)和P(bluue)是否是脱节事件,请列出P(red)和P(bluue)的结果清单:

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  $\begin{align*}R\end{align*}$ outcomes: red
  ::R 结果R:红色

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  $\begin{align*}B\end{align*}$ outcomes: blue
  ::B 结果:蓝色

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  Next, compare the list to decide if the events have common outcomes. If there are no outcomes in common, the two events are disjoint.
  ::其次,比较列表以决定事件是否具有共同结果。 如果没有共同结果,则两个事件是脱节的。

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  Then, if there are no outcomes in common, the two events are disjoint. 
  ::那么,如果没有共同的结果,这两个事件就脱节了。

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  The answer is  $\begin{align*}R\end{align*}$ and $\begin{align*}B\end{align*}$ are disjoint events because they have nothing in common.
  ::答案是R和B是脱节事件,因为它们没有任何共同之处。

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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 Elizabeth and her pirouettes.
  ::早些时候,你得到一个问题 关于伊丽莎白 和她的毛巾。

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  She had accomplished 25 pirouette attempts out of 30 during practice.  Are the two events, performing a pirouette and walking around the block, disjoint events?  Are they mutually exclusive events?
  ::在实践期间,在30次尝试中,她完成了25次圆柱式尝试,这两场活动是执行圆柱式手术,在街区周围走动,是脱节活动吗?它们是否相互排斥?

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  First, to see if  $\begin{align*}P (\text{pirouettes})\end{align*}$  and  $\begin{align*}P (\text{walking around the block})\end{align*}$  are disjoint events, make a list of the outcomes:
  ::首先,看看P(pirouettes)和P(环绕街区行走)是否是脱节事件,

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  $\begin{align*}pirouettes\end{align*}$  outcomes:  pirouettes
  ::转折结果:转折结果

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  $\begin{align*}walking\ around\ the\ block\end{align*}$  outcomes: walking around the block
  ::环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环环

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  Next, compare the list to decide if the events have common outcomes. If there are no outcomes in common, the two events are disjoint.
  ::其次,比较列表以决定事件是否具有共同结果。 如果没有共同结果,则两个事件是脱节的。

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  Then, if there are no outcomes in common, the two events are disjoint. 
  ::那么,如果没有共同的结果,这两个事件就脱节了。

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  The answer is performing pirouettes and walking around the block are disjoint events because they have nothing in common.
  ::答案是使用绕行器,在街区周围行走是脱节事件,因为它们没有任何共同之处。

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  Example 2
  ::例2

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  Are these disjoint events? Explain your thinking.
  ::这些是脱节事件吗?

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  It is going to rain today. I enjoy sailing.
  ::今天要下雨了,我喜欢航海

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  First, to see if  $\begin{align*}P (\text{rain})\end{align*}$  and  $\begin{align*}P (\text{sailing})\end{align*}$  are disjoint events, make a list of the outcomes:
  ::首先,为了看P(rain)和P(配对)是否是脱节事件,请列出结果清单:

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  $\begin{align*}rain\end{align*}$  outcomes: rain
  ::雨水结果:雨

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  $\begin{align*}sailing\end{align*}$  outcomes: sailing
  ::航行结果:航行

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  Next, compare the list to decide if the events have common outcomes.
  ::其次,比较清单,以确定事件是否有共同的结果。

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  Then, if there are no outcomes in common, the two events are disjoint.
  ::那么,如果没有共同的结果,这两个事件就脱节了。

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  The answer is these are disjoint events.  There isn't a connection between it raining and the fact that the person, "I", likes sailing.  
  ::答案是,这些是脱节事件。雨下与“我”喜欢航行之间没有联系。

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  Determine whether the following events are disjoint events or not.
  ::确定以下事件是否为脱节事件。

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  Example 3
  ::例3

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  At a baseball game, hitting a ball and scoring a home run.
  ::在棒球比赛中 打球打全垒打全垒打

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  First, to see if  $\begin{align*}P (\text{hitting a ball})\end{align*}$  and  $\begin{align*}P (\text{scoring a home run})\end{align*}$  are disjoint events, make a list of the outcomes:
  ::首先, 看看 P( 点击球) 和 P( 标注主跑) 是否是脱节事件, 请列出结果列表 :

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  $\begin{align*}hitting\ a\ ball\end{align*}$  outcomes: foul, out, home run 
  ::弹球结果: 犯规、出局、全垒打

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  $\begin{align*}home\ run\end{align*}$  outcomes: home run
  ::主运行结果:主运行

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  Next, compare the list to decide if the events have common outcomes.
  ::其次,比较清单,以确定事件是否有共同的结果。

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  Then, if there are no outcomes in common, the two events are disjoint.  If there are outcomes in common, the two events are not disjoint.
  ::那么,如果没有共同的结果,这两个事件就会分崩离析。 如果有共同的结果,这两个事件就不会分崩离析。

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  The answer is these are not disjoint events.  The event of hitting a ball can result in a home run.  Therefore, the events have common possible outcomes.
  ::答案是,这些事件并非脱节事件。 打球事件可能导致全垒打。因此,这些事件有共同的可能结果。

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  Example 4
  ::例4

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  Eating an ice cream cone and the sun shining
  ::吃冰淇淋甜甜筒 阳光照耀

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  First, to see if  $\begin{align*}P (\text{eating ice cream cone})\end{align*}$  and  $\begin{align*}P (\text{sun shining})\end{align*}$  are disjoint events, make a list of the outcomes:
  ::首先,看看P(吃冰淇淋锥)和P(阳光闪耀)是否是脱节事件,

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  $\begin{align*}eating\ ice\ cream\ cone\end{align*}$  outcomes:  eating ice cream cone
  ::冰淇淋果: 吃冰淇淋果

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  $\begin{align*}sun\ shining\end{align*}$  outcomes: sun shining
  ::太阳耀光结果:太阳耀光

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  Next, compare the list to decide if the events have common outcomes.
  ::其次,比较清单,以确定事件是否有共同的结果。

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  Then, if there are no outcomes in common, the two events are disjoint.
  ::那么,如果没有共同的结果,这两个事件就脱节了。

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  The answer is these are disjoint events.  There isn't a direct connection between eating ice cream and the sun shining.  Either can occur without the other.  
  ::答案是,这些是脱节事件。吃冰淇淋和晒太阳之间没有直接的联系。两者都有可能发生。

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  Example 5
  ::例5

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  Wearing sneakers and running a marathon
  ::穿运动鞋和跑马拉松

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  First, to see if  $\begin{align*}P (\text{sneakers})\end{align*}$  and  $\begin{align*}P (\text{running})\end{align*}$  are disjoint events, make a list of the outcomes:   
  ::首先,看看P(sneakers)和P(running)是否是脱节事件,

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  $\begin{align*}sneakers\end{align*}$  outcomes: performing athetics such as tennis or running  
  ::运动鞋结果:执行诸如网球或跑步等神学

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  $\begin{align*}running\end{align*}$  outcomes: running, as in a marathon
  ::运行结果:运行,如马拉松

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  Next, compare the list to decide if the events have common outcomes.
  ::其次,比较清单,以确定事件是否有共同的结果。

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  Then, if there are no outcomes in common, the two events are disjoint.  If there is a commonality, the two events are not disjoint.
  ::那么,如果没有共同的结果,这两个事件就会分崩离析。 如果有共同之处,这两个事件就不会分解。

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  The answer is these are not disjoint events.  There is a connection between wearing sneakers and running a marathon.
  ::答案是,这些不是脱节事件。穿着运动鞋和跑马拉松之间有联系。

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  Review
  ::回顾

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  Use what you have learned to solve each problem.
  ::利用你学到的东西解决每个问题

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  1. For a single toss of a number cube, are events $\begin{align*}F (\text{four})\end{align*}$ and $\begin{align*}E (\text{even})\end{align*}$ disjoint events or overlapping events?
     ::对于一个数字立方体的一转方块,事件F(四)和E(偶)是脱节事件还是重叠事件?
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  2. For a single toss of a number cube, are $\begin{align*}T (\text{three})\end{align*}$ and $\begin{align*}E (\text{even})\end{align*}$ disjoint events or overlapping events?
     ::对于一个数字立方体的一转方块, T( 3) 和 E( e( even) 是脱节事件还是重叠事件?
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  3. For a single toss of a number cube, are $\begin{align*}O (\text{odd})\end{align*}$ and $\begin{align*}E (\text{even})\end{align*}$ disjoint events or overlapping events?
     ::对于一个数字立方体的一整块,O(odd)和E(e)(偶)是脱节事件还是重叠事件?
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  4. For a single toss of a number cube, are $\begin{align*}T (\text{two})\end{align*}$ and $\begin{align*}L5\end{align*}$ (less than 5) disjoint events or overlapping events?
     ::对于一个数字立方体的单次振标,T(2)和L5(少于5)是否是脱节事件或重叠事件?
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  5. For a single toss of a number cube, are $\begin{align*}S (\text{six})\end{align*}$ and $\begin{align*}O (\text{one})\end{align*}$ disjoint events or overlapping events?
     ::对于一个数字立方体的一方向, S( 6) 和 O( 1) 是脱节事件还是重叠事件?
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  6. For a single toss of a number cube, are $\begin{align*}F (\text{five})\end{align*}$ and $\begin{align*}L5\end{align*}$ (less than 5) disjoint events or overlapping events?
     ::对于一个数字立方体的一转方块,F(5)和L5(少于5)是否断绝了关联事件或重叠事件?

  

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  7. For a single spin, are $\begin{align*}R (\text{red})\end{align*}$ and $\begin{align*}Y (\text{yellow})\end{align*}$ disjoint events or overlapping events?
     ::对于一个旋转, R( red) 和 Y( Y( yellow) 是脱节事件还是重叠事件?
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  8. For a single spin, are $\begin{align*}R (\text{red})\end{align*}$ and $\begin{align*}L (\text{left})\end{align*}$ disjoint events or overlapping events?
     ::就一个旋转而言, R( red) 和 L( left) 是脱节事件还是重叠事件?
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  9. For a single spin, are $\begin{align*}P (\text{yellow})\end{align*}$ and $\begin{align*}P (\text{right})\end{align*}$ disjoint events or overlapping events?
     ::就一个旋转而言,P(yellow)和P(right)是否是脱节事件或重叠事件?
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  10. For a single spin, are $\begin{align*}R (\text{right})\end{align*}$ and $\begin{align*}L (\text{left})\end{align*}$ disjoint events or overlapping events?
      ::就一个旋转而言, R( Right) 和 L( left) 是脱节事件还是重叠事件?
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  11. For a single spin, are $\begin{align*}L (\text{left})\end{align*}$ and $\begin{align*}G (\text{green})\end{align*}$ disjoint events or overlapping events?
      ::对于一个旋转, L( 左) 和 G( 绿色) 事件或重叠事件是否脱节 ?
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  12. For a baby, are $\begin{align*}B (\text{boy})\end{align*}$ and $\begin{align*}R (\text{right-handed})\end{align*}$ disjoint events or overlapping events?
      ::对于婴儿来说,B(男孩)和R(右手)是脱节事件还是重叠事件?
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  13. For a baby, are $\begin{align*}L (\text{left-handed})\end{align*}$ and $\begin{align*}R (\text{right-handed})\end{align*}$ disjoint events or overlapping events?
      ::对于婴儿来说,L(左撇子)和R(右撇子)是否是脱节事件或重叠事件?
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  14. For a baby, are $\begin{align*}G (\text{girl})\end{align*}$ and $\begin{align*}B (\text{brown hair})\end{align*}$ disjoint events or overlapping events?
      ::对于婴儿来说,G(女孩)和B(棕发)是脱节事件还是重叠事件?
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  15. For a baby, are $\begin{align*}G (\text{girl})\end{align*}$ and $\begin{align*}H (\text{heavier than 8 pounds})\end{align*}$ disjoint events or overlapping events?
      ::对于婴儿来说,G(女孩)和H(超过8磅重)是脱节事件还是重叠事件?

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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