12.16 独立活动

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  独立活动

  

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  Miranda is on the school tennis team and she is waiting for the medals to be given out after the district tournament. There are three medals - 1st, 2nd, and 3rd place - that are awarded one at a time. She is standing with two of her teammates and there are five players from other schools.  The 3rd and 2nd place medals are given to the players from other schools.   Is the awarding of the medals an independent event ? 
  ::米兰达在学校网球队中,她正在等待区锦标赛结束后颁发奖牌。有三枚奖牌,一、二和三等奖牌,一、一、二和三等奖牌,一、二、三等奖牌,一、二、三等奖牌,一、二、二、三等奖牌,一、三、三等奖牌,三、二等奖牌,三、二等奖牌,三、三等奖牌,三、三等奖牌,三、二等奖牌,三、三等奖牌,三、三等奖牌奖牌,三、三等奖牌,三等奖牌奖牌,三、三等奖牌,三等奖牌,三等奖牌,三等奖牌,三等奖牌,三、三等奖牌,三、三等奖牌,三等奖牌奖牌,三、三、三等奖牌奖牌,三、三等奖牌奖牌,三、三等奖牌,三等奖牌,三、三等奖牌,三等奖牌获奖牌,三、三等奖牌是独立赛,三等奖牌,三等奖牌,三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三等奖牌奖牌,三、三、三、三等奖牌奖牌,三等奖牌奖牌奖牌奖牌奖牌奖牌,三、三、三、三、三等奖牌奖牌奖牌奖牌奖奖奖奖奖奖奖奖奖奖奖奖奖奖奖,三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、三、

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  In this concept, you will learn about .
  ::在这一概念中,你会知道。

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  Independent Events
  ::独立活动

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  If the outcome of one event has no effect on the outcome of a second event, then the two events are independent events.
  ::如果一个事件的结果对第二个事件的结果没有影响,那么这两个事件是独立的事件。

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  Suppose you have two events:
  ::假设你有两个事件:

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  Event $\begin{array}{r}A\end{array}$ : Spin red on spinner $\begin{array}{r}A\end{array}$
  ::事件A:在旋轴A上旋转红色

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  Event $\begin{array}{r}B\end{array}$ : Spin purple on spinner $\begin{array}{r}B\end{array}$
  ::事件B:在旋盘B上旋转紫色

  

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  The probability of these events is easy enough to compute. In general:
  ::这些事件的概率很容易计算。一般而言:

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  $$\begin{array}{r}P(\text{event})=\frac{\text{favorable outcomes}}{\text{total outcomes}}\end{array}$$

  
  ::P(活动)=有利成果

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  So:
  ::因此:

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  $$\begin{array}{rl}P(\text{red})& =\frac{1}{4}\\ P(\text{purple})& =\frac{1}{3}\end{array}$$

  
  ::P(red) = 14P( purple) = 13

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  Now a question arises:  Does event $\begin{array}{r}A\end{array}$ affect the probability of event $\begin{array}{r}B\end{array}$ in any way? That is, does the arrow landing on red in the first spinner affect the way the arrow lands in the second spinner? If not, then the two events are said to be independent events. 
  ::现在出现一个问题:事件 A 是否以任何方式影响事件 B 的概率? 也就是说, 第一个旋轴中箭头在红色上降落是否会影响箭头在第二个旋轴中降落的方式? 如果没有, 那么这两个事件据说是独立的事件 。

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  Events $\begin{array}{r}A\end{array}$ and $\begin{array}{r}B\end{array}$ above are independent events. No matter how many times you spin spinner $\begin{array}{r}A\end{array}$ , the outcome of spinning spinner $\begin{array}{r}A\end{array}$ does not affect the outcome of spinning spinner $\begin{array}{r}B\end{array}$ .
  ::以上事件 A 和 B 是独立事件。 无论您旋转旋转器 A 多少次, 旋转旋转器 A 的结果都不会影响旋转器 B 的结果 。

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

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  Jeremy flips a coin two times. Event $\begin{array}{r}A\end{array}$ is the first coin flip. Event $\begin{array}{r}B\end{array}$ is the second coin flip. Are the two coin flips independent events?  Can the outcome of the first event in any way change the outcome of the second event? If not, then the two events are independent.
  ::Jeremy翻了两次硬币。事件A是第一个硬币翻了两次。事件B是第二个硬币翻了两次。事件B是第二个硬币翻了两次。两个硬币翻了独立事件吗?第一个事件的结果能以任何方式改变第二个事件的结果吗?如果不是,那么这两个事件是独立的。

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  Suppose Jeremy’s first flip comes up heads. Does that in any way affect the outcome of the second flip? Is it now more likely to come up heads or tails?  In fact, the first flip does not affect the second flip. The probability of heads in the second flip is $\begin{array}{r}\frac{1}{2}\end{array}$ , no matter what the first flip was. Similarly, the probability of tails in the second flip is also $\begin{array}{r}\frac{1}{2}\end{array}$ , no matter what the first flip was. So the two events are independent.
  ::假设Jeremy的第一次翻过头来。 这是否以任何方式影响到第二次翻过的结果?它现在更可能出现头来还是尾来? 事实上,第一次翻过也不影响第二次翻过头去。 第二次翻过头去的概率是12个,不管第一次翻过头去的概率是多少。 同样,第二次翻过尾去的概率也是12个,不管第一次翻过头去的概率是多少。 因此这两个事件是独立的。

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  Here's another example.
  ::下面是另一个例子。

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  Mariko pulls a red sock from the laundry bag. Does this change the probability that the next sock Mariko pulls out of the bag will be red?
  ::Mariko从洗衣袋里拉了一只红袜子。这是否改变了下一个袜子Mariko从袋子里拉出来的可能性?

  

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  Here, the act of taking a sock out of the bag changes the situation. For the first sock, the probability of pulling out a red sock was:
  ::在这里,把袜子从袋子里拿出来的行为改变了情况。对于第一只袜子,拉出红袜子的概率是:

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  $$\begin{array}{r}P(\text{red})=\frac{\text{favorable outcomes}}{\text{total outcomes}}=\frac{3}{6}=\frac{1}{2}\end{array}$$

  
  ::P(red)=有利成果

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  For the second sock, there are now only 5 socks left in the bag and only 2 of them are red. So the probability of pulling out a red sock now for the second sock is:
  ::第二只袜子现在只剩下5只袜子了,其中只有2只是红色的。因此,现在第二只袜子可能拔出红袜子的概率是:

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  $$\begin{array}{r}P(\text{red})=\frac{\text{favorable outcomes}}{\text{total outcomes}}=\frac{2}{5}\end{array}$$

  
  ::P(red)=有利成果

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  Clearly, the first event affected the outcome of the second event in this situation. So the two events are NOT independent. 
  ::显然,第一个事件影响了第二个事件在这种情况下的结果。 因此,这两个事件并不独立。

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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 Miranda, who was waiting to find out if she would receive the 1st-place medal.
  ::早些时候,有人给了你一个问题 关于米兰达, 谁在等待 想知道她是否会得到第一名奖牌。

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  There are three medals that are awarded one at a time for 1st place, 2nd place, and 3rd place.  The 2nd and 3rd-place medals have already been awarded to players from another school.  Is the awarding of the 1st-place medal an independent event?
  ::第一、第二和第三位分别获得三个奖牌。 第二和第三位的奖牌已经授予另一所学校的球员。 第一和第三位的奖牌是独立活动吗?

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  First, evaluate the 1st and 2nd outcomes :
  ::首先,评价第一和第二项成果:

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  The 3rd-place medal was awarded to a player from the other school.
  ::第三名奖牌授予了另一所学校的一位球员。

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  The 2nd-place medal was then awarded to another player from the other school.
  ::第二名奖牌随后颁发给另一所学校的另一位运动员。

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  Next, evaluate the 3rd outcome:
  ::接下来,评价第三个结果:

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  The 1st-place medal will be awarded to one of the remaining players, Miranda, one of her teammates, or the three remaining players from the other school.
  ::第一位奖章将颁发给剩下的一名球员,米兰达,她的队友之一,或来自另一所学校的其余三名球员。

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  Then, decide if the 1st and 2nd outcomes affect the 3rd outcome:
  ::然后决定第1和第2项结果是否影响第3项结果:

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  Each time a medal is presented, it is no longer available to the remaining players.  There is only 1 medal left for the 3rd outcome.
  ::每颁发一枚奖牌,就不再可供其余的球员使用。 只有一枚奖牌可供第三次比赛使用。

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  The answer is the events are not independent.
  ::答案是事件不是独立的。

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  Example 2
  ::例2

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  Does the following situation describe independent events?
  ::以下情况是否描述独立事件?

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  Kelsey has a drawer full of earrings and necklaces. She has four pairs of earrings and six necklaces in the drawer. Kelsey first takes out a necklace. She isn't happy with it, so she puts it back into the drawer. What is the probability of pulling another necklace?
  ::Kelsey的抽屉里装满了耳环和项链,她有四对耳环和六条项链。Kelsey首先取出一条项链,她不满意,所以她把它放回抽屉里。再拉一条项链的概率是多少?

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  First, evaluate the 1st outcome:
  ::首先,评价第一个结果:

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  Kelsey takes a necklace.  She replaces it. 
  ::Kelsey拿了项链 她换了项链

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  Next, evaluate the 2nd outcome:
  ::接下来,评价第二个结果:

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  Kelsey will be taking a piece of jewelry from the same collection as she did for the 1st outccome.
  ::Kelsey会拿一块珠宝 从相同的收藏品 与她为第一胜出所做的一样。

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  Then, decide if the 1st outcome affects the 2nd outcome:
  ::然后,决定第一个结果是否影响第二个结果:

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  Since Kelsey replaced the necklace she took, the 1st outcome does not affect the 2nd outcome.
  ::由于凯尔西取代了她拿走的项链, 第一项结果不影响第二项结果。

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  The answer is the two events are independent.
  ::答案是这两个事件是独立的。

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  Determine whether or not the events described are independent events.
  ::确定所述事件是否为独立事件。

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  Example 3
  ::例3

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  A box contains a penny, a nickel, a dime, and a quarter. What is the probability of pulling a quarter out of the box, putting it in your pocket, then pulling a penny out of the box?
  ::一个盒子里装着一便士,一便士,一便士,一便士,一便士,一便士。从盒子里拉出四分之一,放在口袋里,然后从盒子里拉出一便士的概率是多少?

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  First, evaluate the 1st outcome:
  ::首先,评价第一个结果:

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  You pull a quarter out of a box and placed in your pocket. 
  ::你从盒子里拿出25美分 放进口袋里

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  Next, evaluate the 2nd outcome:
  ::接下来,评价第二个结果:

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  You will pull another coin from the box.
  ::你会从盒子里拿出另一枚硬币

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  Then, decide if the 1st outcome affects the 2nd outcome:
  ::然后,决定第一个结果是否影响第二个结果:

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  Since you did not place the quarter back into the box, there is one less coin to pull from for the 2nd outcome
  ::既然你没有把季度再放回盒子里 就少了一枚硬币 来换取第二个结果

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  The answer is the two events are not independent.  The 1st coin pull affects the outcome of the 2nd coin pull.
  ::答案是这两个事件并不独立。 第一次硬币拉动会影响第二次硬币拉动的结果。

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  Example 4
  ::例4

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  In a laundry bag with 3 red socks and 3 blue socks, Mariko pulls out a blue sock, sees it’s the wrong sock and returns it to the bag. Now Mariko pulls out a second sock. What is the probability that it will be red?
  ::Mariko用三只红袜子和三只蓝袜子在洗衣袋里拉出一只蓝袜子,看到错误的袜子,然后把它还给袋子。 现在,Mariko又拔出第二只袜子。 红袜子的红色概率有多大?

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  First, evaluate the 1st outcome:
  ::首先,评价第一个结果:

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  Mariko pulls a blue sock out of the laundry bag and replaces it.
  ::玛里子从洗衣袋里拿出一只蓝袜子 换掉它

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  Next, evaluate the 2nd outcome:
  ::接下来,评价第二个结果:

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  Mariko will pull another sock out of the laundry bag.
  ::玛里子会从洗衣袋里再拉一只袜子出来

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  Then, decide if the 1st outcome affects the 2nd outcome:
  ::然后,决定第一个结果是否影响第二个结果:

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  Since Mariko replaced the sock he pulled out of the laundry bag for the 1st outcome, the same socks remain in the laundry bag for the 2nd outcome.
  ::自从Mariko换掉他从洗衣袋里取出的袜子 第一次结果, 同样的袜子留在洗衣袋里,第二次结果。

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  The answer is the two events are independent.  Taking the sock from the bag for the 1st outcome does not affect taking a sock from the bag for the 2nd outcome since both pulls contain the same socks.
  ::答案是两个事件是独立的。把袜子从袋子中取出以取得第一个结果并不影响把袜子从袋子中取出以取得第二个结果,因为两次抽取都包含相同的袜子。

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  Example 5
  ::例5

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  In a laundry bag with 3 red socks and 3 blue socks, Mariko pulls out a blue sock and keeps it out. Now Mariko pulls out a second sock. What is the probability that it will be blue?
  ::Mariko用三只红袜子和三只蓝袜子在洗衣袋里拉出一只蓝袜子,把它挡住。现在,Mariko又拉出第二只袜子。它的蓝色概率是多少?

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  First, evaluate the 1st outcome:
  ::首先,评价第一个结果:

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  Mariko takes a sock from the laundry bag and keeps it.
  ::玛里子从洗衣袋里拿了一只袜子 留着它

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  Next, evaluate the 2nd outcome:
  ::接下来,评价第二个结果:

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  He will then pull another sock.
  ::然后他再拉另一只袜子

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  Then, decide if the 1st outcome affects the 2nd outcome:
  ::然后,决定第一个结果是否影响第二个结果:

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  Since Mariko did not replace the sock on the 1st pull, there is one less sock to pull from on the 2nd pull. 
  ::因为Mariko没有取代第一次拉动时的袜子,因此从第二次拉动时的袜子少了一个。

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  The answer is the two events are not independent because there is one less sock to pull from on the 2nd pull.
  ::答案是这两个事件并不独立, 因为有一个袜子少了,

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  Review
  ::回顾

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  Write whether each pair of events is dependent or independent.
  ::写下每一对事件是依附还是独立的。

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  1. A: Mike rolls a number cube. B: Leah spins a red-blue-green spinner.
     ::答:麦克滚动了一个数字立方体。B:莉亚旋转了一个红色蓝色绿色的旋转器。
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  2. A: In a game of Go Fish, the probability of one player drawing a Queen from the deck. B: On the next player’s turn, the probability of drawing a Queen.
     ::甲:在Go Fish游戏中,一个玩家从甲板上画皇后的概率。乙:下一个玩家轮到谁,就画皇后的概率。
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  3. A: The probability that a randomly ordered pizza will be large. B: The probability that the same randomly ordered pizza will be deep-dish.
     ::A: 随机订购的比萨饼的概率会很大。 B: 随机订购的比萨饼的概率也会大得多。 B: 随机订购的比萨饼的概率也会大得多。
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  4. A: The probability that a randomly ordered 2-topping pizza will have pepperoni. B: The probability that the same randomly ordered 2-topping pizza will have mushrooms.
     ::A: 随机订购的2吨披萨有辣椒的概率。 B: 同样随机订购的2吨披萨有蘑菇的概率。 B: 随机订购的2吨披萨有蘑菇的概率。
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  5. A: The probability of flipping a coin tails 5 times in a row. B: The probability of the sixth flip turning out to be heads.
     ::A: 翻翻硬币尾巴5次的概率。 B: 第6次翻转的概率是头。
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  6. A: In a 4-team league, the probability of the Rockets finishing in first place. B: In a 4-team league, the probability of the Sharks finishing in first place.
     ::A:在一个四队联盟中,火箭的完成概率居第一位。B:在四队联盟中,鲨鱼的完成概率居第一位。
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  7. A: On a roll of a number cube, the probability of rolling 6. B: On a second roll of a number cube, the probability of rolling 6.
     ::A:在数字立方体的滚动中,滚动的概率为6.B:在数字立方体的第二个滚动中,滚动的概率为6。
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  8. A: In a spelling bee, the probability of the first contestant being given the word khaki from a list of 10 words. B: In a spelling bee, the probability of the second contestant getting the word khaki from the same list of words.
     ::A:在拼写蜜蜂中,第一个参赛者从10个单词列表中获得卡其字的概率。B:在拼写蜜蜂中,第二个参赛者从相同的单词列表中获得卡其字的概率。
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  9. A: The probability that it will snow on Tuesday. B: The probability that Tuesday will fall on an odd day of the month.
     ::A: 星期二下雪的概率。 B: 星期二下雪的概率 。 B: 星期二下雪的概率
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  10. A: The probability that it will be below 32 degrees on Tuesday. B: The probability that it will snow on Tuesday.
      ::A: 星期二的降雪概率低于32度。 B: 星期二下雪的概率。
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  11. A: The probability that it will snow on Tuesday. B: The probability that school will be cancelled on Tuesday.
      ::A: 星期二下雪的概率。 B: 星期二学校取消的概率。
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  12. A: The probability that the first Wednesday in June will fall on an even day of the month. B: The probability that the first Thursday in June will fall on an even day of the month.
      ::A: 6月第一个星期三在月中一个偶数日的概率。 B: 6月第一个星期四在月中一个偶数日的概率。 B: 6月第一个星期四在月中一个偶数日的概率。
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  13. A: The probability that the first Wednesday in June will fall on an even day of the month. B: The probability that the first Thursday in June will be sunny.
      ::A: 6月第一个星期三的几率将在每月的某一天降下。 B: 6月第一个星期四的几率将是晴天。
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  14. A: The probability that a coin will land on heads. B: The probability that a number cube will land on 5.
      ::A:硬币落到头顶的概率。B:数立方体落到5的概率。
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  15. A: The probability that the first spin of a red-blue-green spinner will land on green. B: The probability that the second spin of a red-blue-green spinner will land on green.
      ::A: 红色蓝绿色旋转器的第一个旋转器在绿色上着陆的概率。 B: 红色蓝绿色旋转器第二个旋转器在绿色上着陆的概率。 B: 红色蓝绿色旋转器第二个旋转器在绿色上着陆的概率 。

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  Review (Answers)
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