12.5 相互包容的活动

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  相互包容的活动

  

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  A magician at a magic show says he will throw a deck of cards in the air and pierce a red card, either hearts or diamonds, or a 9.  Is this an example of a mutually inclusive event ?  What is the probability of one of these events occurring?
  ::魔术师在魔术表演中说,他要把一副牌扔到空中,穿穿红卡片,红卡片或红卡片,或红卡片,或红卡片,或红卡片,或钻石,或9红卡片,或9红卡片,或者红卡片,或者红卡片,或者红卡片,或者红卡片,或者红卡片,或者钻石,或者9卡片,或者9卡片,这是相互包容事件的例子吗?发生这些事件的概率是多少?

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  In this concept, you will learn how to calculate the probability of overlapping, or mutually inclusive events. 
  ::在此概念中,您将学会如何计算重叠或相互包容事件的概率。

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  Mutually Inclusive Events
  ::相互包容的活动

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  There are some events that can impact each other or that are overlapping. They have one or more outcomes in common. When events have outcomes in common, they are said to be  overlapping events , also known as  mutually inclusive events . 
  ::有些事件可以相互影响或相互重叠,它们有一个或多个共同结果,当事件结果相同时,据说是重叠事件,也称为相互包容的活动。

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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 $\begin{array}{r}R(\text{red})\end{array}$ and $\begin{array}{r}T(\text{top})\end{array}$  overlapping events?
  ::单一个旋转,事件 R( red) 和 T( top) 是重叠事件吗 ?

  

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  First, make a list of the outcomes.
  ::首先,列出一个结果清单。

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  $\begin{array}{r}R\end{array}$ outcomes: red-top, red bottom
  ::R 结果: 红下, 红上, 红下

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  $\begin{array}{r}T\end{array}$ outcomes: red-top, blue-top
  ::T 结果:红色、蓝色

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  Next, compare the list.
  ::接下来,比较一下清单。

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  Then, decide if the two events in question have a common outcome .
  ::然后,决定这两个事件是否有共同的结果。

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  The two events DO have an outcome in common–red-top. 
  ::这两次事件的结果是 共同的 - - - 红顶。

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  The answer is the two events do have a common outcome, therefore  $\begin{array}{r}R\end{array}$ and $\begin{array}{r}T\end{array}$ are mutually inclusive events.
  ::答案是,这两个事件确实有着共同的结果,因此R和T是相互包容的活动。

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  So  $\begin{array}{r}R\end{array}$ and $\begin{array}{r}T\end{array}$ are mutually inclusive events .   Notice that the events have more than one thing in common. They have red in common, but they also have the words “top”  in common too. 
  ::所以R和T是相互包容的事件。请注意,这些事件有不止一个共同点。它们有红色的共同点,但它们也有“最上面”的共通点。

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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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  For a single toss of a number cube, are events smaller than 6 and greater than 4 disjoint events or overlapping events?
  ::对于一个数字立方体的一转方块,事件小于6或大于4个脱节事件或重叠事件吗?

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  First, make a list of the outcomes.
  ::首先,列出一个结果清单。

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  Smaller outcomes 1, 2, 3, 4, 5, 
  ::较小结果1、2、3、4、5、

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  Greater outcomes: 5, 6
  ::更大成果:5、6

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  Next, compare the list.
  ::接下来,比较一下清单。

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  Then, decide if the two events in question have a common outcome.
  ::然后,决定这两个事件是否有共同的结果。

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  The two events have 1 outcome in common = 5.
  ::这两个事件有一个共同结果=5。

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  The answer is events smaller than 6,  $\begin{array}{r}S6\end{array}$ , and greater than 4,   $\begin{array}{r}G4\end{array}$ , are mutually inclusive events.
  ::答案是小于6个S6的事件,超过4个G4的事件是相互包容的事件。

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  To find the probability of mutually inclusive events occurring, the following formula is used:
  ::为了发现发生相互包容事件的可能性,采用了以下公式:

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  X = one set of outcomes   Y = another set of outcomes, where X and Y are known to have mutually inclusive events:
  ::X = 一组结果Y = 另一组结果,据知X 和Y 具有相互包容的活动:

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  $\begin{array}{r}P(\text{A or}B)=P(A)+P(B)-P(AandB)\end{array}$
  ::P(A或B)=P(A)+P(B)-P(A和B)

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  where P(A or B) is the probability tht either event A or event B will occur, P(A) is the probability that event A will occur, P(B) is the probability that event B will occur and P(A and B) is the probability that both events will occur.
  ::P(A或B)是事件A或事件B的概率。 P(A)是事件A发生的概率。 P(A)是事件A发生的概率。 P(B)是事件B发生的概率。 P(A和B)是两种事件发生的概率。

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  For example:
  ::例如:

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  A number cube is tossed. What is the probability that it will be an even number or a number greater than four?
  ::一个数字立方体被丢弃。它是一个偶数或数大于4的概率是多少?

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  even outcomes:  2, 4, 6
  ::偶等结果:2、4、6

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  >  4 outcomes:  5, 6
  ::> 4项成果:5、6

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  When you consider the outcomes, the mutually inclusive outcome, the one that both events have in common, is 6.
  ::当你考虑结果时, 相互包容的结果, 两个事件的共同点是6。

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    $\begin{array}{r}P(\text{even outcomes or}greaterthan4)=P(\frac{3}{6})+P(\frac{2}{6})-P(\frac{1}{6})\end{array}$
  ::P(偶数结果或大于4)=P(36)+P(26)-P(16)

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  The answer is the probability that a number cube will land on an even number or a number greater than four is  $\begin{array}{r}\frac{4}{6}=\frac{2}{3}=0.67=67percent\end{array}$ .
  ::答案是数字立方体在偶数或大于4的数值上着陆的概率是 46=23=0.67=67%。

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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  the magician and his promise.
  ::之前,你被问及魔术师和他的承诺

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  He claims he will pierce a red or a 9 out of a deck of cards thrown up into the air.  Is this a mutually inclusive event and, if so, what is the probability one of the favorable events will occur?
  ::他声称,他将穿透一张红色或9张被扔到空中的牌甲板上的红色或9张牌。 这是相互包容的事件吗? 如果是这样,那么有利事件的概率是什么?

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  First, make list of all outcomes:
  ::首先,列出所有结果:

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  red cards in a deck outcomes: 1 hearts 1 diamonds, 2, hearts, 2 diamonds, 3 hearts, 3 diamonds, 4, hearts, 4 diamonds, 5 hearts, 5 diamonds, 6 hearts, 6 diamonds, 7 hearts, 7 diamonds, 8 hearts, 8 diamonds, 9 hearts, 9 diamonds, 10 hearts, 10 diamonds, Jack hearts, Jack diamonds, Queen hearts, Queen diamonds, King hearts, King diamonds, Ace heart, Ace diamonds
  ::红卡牌在甲板结果中:1个红卡,1个红卡1个钻石、2个红卡、2个钻石、3个红卡、3个钻石、4个红卡、4个钻石、5个红卡、5个钻石、5个钻石、6个红卡、6个钻石、7个红卡、7个钻石、8个红卡、8个钻石、9个红卡、9个钻石、10个红卡、10个钻石、杰克红卡、杰克钻石、杰克钻石、皇后红卡、皇后钻石、国王红金、国王钻石、Ace心脏、Ace钻石

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  9 cards in a deck:  9 clubs, 9 spades, 9 diamonds, 9 hearts
  ::甲板9张牌:9个俱乐部,9个黑桃,9个钻石,9个红心

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  Next, consider the list and identify if any mutuallly inclusive events are present:
  ::下一步,审议清单,并确定是否存在任何相互包容的活动:

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  There are 2 mutuallly inclusive events:  9 diamonds, 9 hearts.
  ::共有两个相互包容的活动:9个钻石、9个心脏。

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  Then, create a ratio of favorable outcomes to total outcomes to indicate the probability of the favorable outcome occurring:
  ::然后,设定一个有利结果与总结果的比例,以表明出现有利结果的可能性:

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    $\begin{array}{r}P(\text{9 or}red)=P(\frac{4}{52})+P(\frac{26}{52})-P(\frac{2}{52})=\frac{28}{52}=0.54=54percent\end{array}$
  ::P(9或红色)=P(452)+P(2652)-P(252)=2852=0.54=54%

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  The answer is that the events of a red 9 out of a deck of cards being pierced by the magician is mutually inclusive.  The probability of this outcome is 1/26.
  ::答案是魔术师穿透的牌牌牌牌上红色9的红色9事件是相互包容的,结果的概率是1/26。

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  Example 2
  ::例2

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  A number cube is tossed. What is the probability that the number that lands up will be odd or greater than 3?
  ::数字 立方体被丢弃。 投放的数值会奇数或大于 3 的概率是多少 ?

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  First, list the odd outcomes, outcomes greater than 3, and total outcomes. 
  ::首先,列出奇数结果、超过3的结果和总结果。

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  odd outcomes: 1, 3, 5
  ::奇数结果: 1, 3, 5

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  > 3 outcomes: 4, 5, 6
  ::> 3个结果:4、5、6

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  total outcomes: 1, 2, 3, 4, 5, 6
  ::结果合计:1,2,3,4,4,5,6

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  Next, consider the list and identify the mutually inclusive outcomes:
  ::其次,审议清单并确定相互包容的结果:

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  There is only one mutually inclusive outcome, the favorable outcome, which is 5.
  ::只有一个相互包容的结果,即有利结果,即5。

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  Then, using the formula for mutually inclusive probability, calculate the probability percent:
  ::然后,使用相互包容概率的公式,计算概率百分比:

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  $\begin{array}{r}P(\text{odd outcomes or}greaterthan3)=P(\frac{3}{6})+P(\frac{3}{6})-P(\frac{1}{6})=\frac{5}{6}=0.83=83percent\end{array}$
  ::P(已知结果或大于3)=P(36)+P(36)-P(16)=56=0.83=83%

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  Example 3
  ::例3

  

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  For a single spin of the spinner above, are $\begin{array}{r}G(\text{green})\end{array}$ and $\begin{array}{r}T(\text{top})\end{array}$ disjoint events or overlapping events?
  ::对于上面一个旋转符的单个旋转, G( 绿色) 和 T( 顶) 是脱节事件还是重叠事件 ?

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  First, make a list of the outcomes.
  ::首先,列出一个结果清单。

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  $\begin{array}{r}G(\text{green})\end{array}$  outcomes = green-bottom
  ::G(绿色)结果 = 绿底

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  $\begin{array}{r}T(\text{top})\end{array}$  outcomes = red-top, blue-top
  ::T(顶)结果 = 红顶、蓝顶

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  Next, compare the list.
  ::接下来,比较一下清单。

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  Then, decide if the two events in question have a common outcome.
  ::然后,决定这两个事件是否有共同的结果。

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  The two events do not have a common outcome.
  ::这两个事件没有共同的结果。

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  The answer is $\begin{array}{r}G(\text{green})\end{array}$  and $\begin{array}{r}T(\text{top})\end{array}$   are disjoint events because they do not have any outcomes in common.
  ::答案是G(绿色)和T(顶部)是脱节事件,因为它们没有任何共同的结果。

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  Example 4
  ::例4

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  A number cube is tossed. What is the probability that it will be a two or an even number?
  ::一个数字立方体被丢弃。它是一个两个或一个偶数的概率是多少?

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  First, list all possible outcomes: 
  ::首先,列出所有可能的结果:

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  2 outcomes: 2
  ::2项成果:2项成果:2

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  even outcomes: 2, 4, 6
  ::偶等结果:2、4、6

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  total outcomes: 2, 4, 6
  ::结果共计:2、4、6

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  Next, consider the list and identify the mutually inclusive outcomes:
  ::其次,审议清单并确定相互包容的结果:

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  There is only one mutually inclusive outcome, the favorable outcome, which is 2.
  ::只有一个相互包容的结果,一个有利结果,即2。

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  Then,  create the ratio of the number of favorable outcomes to total outcomes.
  ::然后,建立有利结果与总结果之比。

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    $\begin{array}{r}P(2oreven)=P(\frac{1}{6})+P(\frac{3}{6})-P(\frac{1}{6})=\frac{3}{6}=0.50=50percent\end{array}$
  ::P(2或甚至)=P(16)+P(36)-P(16)=36=0.50=50%

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  The answer is the probability of a number cube landing on a two and an even number is   $\begin{array}{r}\frac{1}{6}\end{array}$ .
  ::答案是数字立方体降落在两个或一个偶数是16的概率。

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  Review
  ::回顾

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  Write the probability of each overlapping event occurring as a fraction. Remember to simplify.
  ::将每个重叠事件的概率写成一个分数。 记住要简化 。

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  Eight colored scarves are put into a bag. They are red, yellow, blue, green, purple, orange, brown and black.
  ::八条彩色的围巾放在包里,是红色、黄色、蓝色、绿色、紫色、橙色、棕色和黑色的。

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  1. What is the probability of pulling out a red scarf?
     ::拔出红色围巾的概率是多少?
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  2. What is the probability of pulling out a blue scarf?
     ::拔出蓝色围巾的概率是多少?
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  3. What is the probability of pulling out a red or blue scarf?
     ::拔出红色或蓝色围巾的概率是多少?
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  4. What is the probability of pulling out a blue or green scarf?
     ::拔出蓝色或绿色围巾的概率是多少?
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  5. What is the probability of pulling out a green or purple scarf?
     ::拔出绿色或紫色围巾的概率是多少?

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  A twelve sided die is rolled.
  ::12个侧面死亡是滚动的。

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  6. What is the probability of rolling an even number?
     ::滚动偶数的概率是多少?
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  7. What is the probability of rolling an odd number?
     ::滚动奇数的概率是多少?
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  8. What is the probability of rolling an even number or a number less than four?
     ::滚动偶数或数字小于4的概率是多少?
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  9. What is the probability of rolling an even number or a number greater than 10?
     ::滚动偶数或数字大于10的概率是多少?
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  10. What is the probability of rolling an odd number or a number greater than 5?
      ::滚动奇数或数字大于5的概率是多少?
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  11. What is the probability or rolling an odd number or a one?
      ::概率或滚动一个奇数或一个数字的概率是多少?
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  12. What is the probability of rolling an odd number or a number less than 8?
      ::滚动奇数或数字小于8的概率是多少?
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  13. What is the probability of rolling an even number or a number less than 11?
      ::滚动偶数或数字小于11的概率是多少?
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  14. What is the probability of rolling an odd number or a number greater than 9?
      ::滚动奇数或数字大于9的概率是多少?

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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 。

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  Resources
  ::资源