12.8 查明事件的概率

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  查找事件的可能性

  

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  You have a standard deck of 52 cards. You draw the top card from the well-shuffled deck. What is the probability your card will be an ace?
  ::您有52张牌的标准牌牌。 您可以从牌堆中抽取牌顶牌。 您的牌有A的概率吗 ?

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  Finding the Probability of an Event
  ::查找事件的可能性

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  The probability of a particular outcome of an event occurring is a measure of how likely the desired outcome is to occur. In this concept we will calculate the probability of an event using the ratio of the number of ways the desired outcome can occur to the number of items in the sample space . The sample space is essentially a list of all the possible outcomes . For example, when we roll a single die, the sample space is $\begin{align*}\{1, 2, 3, 4, 5, 6\}\end{align*}$ because these are all the possible outcomes. So, the probability of rolling a 3 is $\begin{align*}\frac{1}{6}\end{align*}$ because there is one way to roll a 3 and there are 6 elements in the sample space. We can write a rule for probability when all the outcomes in the sample space are equally likely:
  ::发生事件特定结果的概率是衡量发生预期结果的可能性的尺度。 在这个概念中, 我们将使用所希望的结果与样本空间中项目数量的比例来计算事件的概率。 样本空间基本上是所有可能结果的列表。 例如, 当我们滚动一次死亡时, 样本空间是 {1, 2, 3,4, 5,6}, 因为这些都是可能的结果。 因此, 滚动一个 3 的概率是 16 , 因为有一个方法可以滚动一个 3 , 在样本空间里有 6 个元素。 我们可以写出一个概率规则, 当样本空间中的所有结果都同样可能发生时 :

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  $$\begin{align*}P(\text{event}) = \frac{\text{number of desirable outcomes}}{\text{number of outcomes in the sample space}}\end{align*}$$
  ::P(活动)=样本空间结果的预期结果数目

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  The problems  below are more accurately described as theoretical probabilities because the theory is that if all the outcomes have an equal likelihood of occurring then the probability is the ratio described about. Does this mean however that when we flip a coin four times we will get 2 heads and 2 tails? Theoretically, this is the most likely outcome, but it is possible that in an experiment we get very different results. When we use the results of an experiment to determine probabilities they are referred to as experimental probabilities. You can complete the following table to investigate the connection between theoretical and experimental probabilities.
  ::下面的问题更准确地被描述为理论上的概率,因为理论是,如果所有结果都具有同样的可能性,那么概率就是所描述的概率。这是否意味着当我们翻硬币时,我们就会得到两个头和两个尾巴。从理论上讲,这是最有可能的结果,但在实验中,我们有可能得到非常不同的结果。当我们使用实验结果来确定概率时,它们被称为实验概率。你可以填写下表来调查理论概率和实验概率之间的联系。

  Number of flips of a coin	Number of Heads	Number of Tails	Probability of flipping a Head	Probability of flipping a Tail
  5
  10
  50
  100
  $\begin{align*}1000^{\ast}\end{align*}$
  Theoretical			$\begin{align*}\frac{1}{2}\end{align*}$	$\begin{align*}\frac{1}{2}\end{align*}$

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  $\begin{align*}^*\end{align*}$ You may wish to use a probability simulator to investigate how many heads and tails are achieved with 1000 flips or combine your results for the 100 flips with 9 other classmates.
  ::* 您可以使用概率模拟器来调查用1000个翻转实现多少头部和尾部,或者将100个翻转的结果与其他9个同学结合起来。

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  Is the experimental probability the same as the theoretical probability ? What do you notice as the number of flips increases?
  ::实验概率是否与理论概率相同?您注意到什么是翻转数的增加?

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  Let's find the probability for the following situations.
  ::让我们找出以下情形的概率。

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  1. What is the probability of rolling a single die and obtaining a prime number?
     ::一次死亡和获得质数的概率是多少?

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   In this problem  there are exactly 3 prime numbers on the die $\begin{align*}\{2, 3, 5\}\end{align*}$ and there are six elements in the sample space $\begin{align*}\{1, 2, 3, 4, 5, 6\}\end{align*}$ so $\begin{align*}P(\text{prime}) = \frac{3}{6}=\frac{1}{2}\end{align*}$ .
  ::在这个问题中,死亡的 {2,3,5} 正好有3个质数,样本空间有6个元素 {1,2,3,4,5,6} 所以P(质数)=36=12。

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  2. What is the probability of rolling a four and a three when two dice are rolled? How about a sum of six when two dice are rolled? What is the most likely sum to roll?
     ::当两个骰子被滚动时滚动一个四,一个三的概率是多少?当两个骰子被滚动时滚动的概率是多少?当两个骰子被滚动时滚动的概率是6,如何?最有可能滚动的数值是多少?

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   In this case it is useful to make a diagram of the sample space when two dice are rolled.
  ::在这种情况下,宜在滚动两个骰子时绘制一个样本空间图。

  $$\begin{align*}& 1,1 \quad 2,1 \quad 3,1 \quad 4,1 \quad {\color{blue}5,1} \quad 6,1 \\ & 1,2 \quad 2,2 \quad 3,2 \quad {\color{blue}4,2} \quad 5,2 \quad 6,2 \\ & 1,3 \quad 2,3 \quad {\color{blue}3,3} \quad {\color{red}4,3} \quad 5,3 \quad 6,3 \\ & 1,4 \quad {\color{blue}2,4} \quad {\color{red}3,4} \quad 4,4 \quad 5,4 \quad 6,4 \\ & {\color{blue}1,5} \quad 2,5 \quad 3,5 \quad 4,5 \quad 5,5 \quad 6,5 \\ & 1,6 \quad 2,6 \quad 3,6 \quad 4,6 \quad 5,6 \quad 6,6\end{align*}$$

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  From this diagram we can see that there are 36 possible outcomes when two dice are rolled.
  ::从这张图表中可以看出,在两个骰子滚动时,可能会有36个结果。

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  • To answer the first part of the question, we can observe in the table that there are two ways (shown in $\begin{align*}{\color{red}\text{red}}\end{align*}$ ) that can we roll a 4 and a 3 so the probability of rolling a 4 and a 3 is $\begin{align*}\frac{2}{36}=\frac{1}{18}\end{align*}$ .
    ::为了回答问题的第一部分,我们可以在表格中看到,有两种方法(红色显示)可以滚4和3,因此滚4和3的概率是236=118。

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  • For the second part, there are 5 ways (shown in $\begin{align*}{\color{blue}\text{blue}}\end{align*}$ ) that a sum of six can be rolled. Therefore the probability of rolling a sum of six is $\begin{align*}\frac{6}{36}=\frac{1}{6}\end{align*}$ .
    ::第二部分有5种方法(以蓝色显示),6个之和可以滚动。因此滚动6个之和的概率是636=16。

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  3. In a case study of an experimental drug, there were 80 participants. Of the 80 participants, 65 of them experienced no significant side effects from the treatment. What is the probability of a person taking the drug to experience significant side effects? How accurate do you think this probability is? Justify your answer.
     ::在试验药物的案例研究中,有80名参与者。在80名参与者中,有65人没有从治疗中得到任何重大的副作用。一个人服用该药物来经历重大副作用的概率是多少?你认为这种概率有多准确?你是否认为这种可能性是多少?你回答的正确性。

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  If 65 of the 80 participants did not experience significant side effects, then 15 of them did. So the likelihood of someone in the future experiencing a significant side effect to the drug is $\begin{align*}\frac{15}{80}=\frac{3}{16}\end{align*}$ . This is experimental probability and as we learned in the investigation, the accuracy of this type of probability will increase as the number of trials in the study increases. Also, as individuals and their general health vary, so will the likelihood of a particular person to experience side effects from a drug vary.
  ::如果在80名参与者中,65人没有经历重大的副作用,那么其中15人没有。因此,某人今后可能遭受药物的显著副作用是1580=316,这是实验性概率,正如我们在调查中了解到的,随着研究试验次数的增加,这种概率的准确性将会增加。此外,由于个人及其一般健康状况不同,某个人遭受药物副作用的可能性也会不同。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to find  the probability that your card will be an ace. 
  ::早些时候,有人要求你 找出你的卡会成为王牌的概率

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  There are 52 possible cards that can be drawn. Each deck contains 4 aces, one of each suit (spades, clubs, hearts, and diamonds).
  ::可能有52张牌可以画,每张甲板有4张A,每套西装有1张(纸牌、俱乐部、红心和钻石)。

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  The probability that the card you draw from the top of the deck is an ace is therefore $\begin{align*}\frac {4}{52} = \frac{1}{13}\end{align*}$ .
  ::从甲板顶部抽取的牌是A的概率是452=113。

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  Example 2
  ::例2

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  What is the probability of selecting a red chip from a bag containing 10 red chips, 12 blue chips and 15 white chips?
  ::从装有10个红薯片、12个蓝薯片和15个白薯片的袋子中选择一个红薯片的概率是多少?

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  $\begin{align*}P(\text{red})=\frac{10}{10+12+15}=\frac{10}{37}\end{align*}$ .
  ::P(red)=1010+12+15=1037。

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  Example 3
  ::例3

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  What is the probability of rolling doubles when two dice are rolled?
  ::当两个骰子被滚动时,滚动双倍的概率是多少?

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  There are six ways to roll doubles $\begin{align*}(1, 1), (2, 2),\end{align*}$ etc (refer back to the diagram in problem #2). So $\begin{align*}P(\text{doubles})=\frac{6}{36}=\frac{1}{6}\end{align*}$ .
  ::有六种方法可以滚动双倍(1,1,2,2),等等(请参考问题2中的图表)。所以P(双倍)=636=16。

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  Example 4
  ::例4

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  Over the course of a month, Sally and Stan recorded how many times their cell phones dropped a call. During this time Sally made 55 phone calls and 4 of them were dropped while Stan made 36 calls and 3 were dropped. What is the probability that Sally’s cell phone drops a call? How about Stan’s? Who appears to have to more reliable service?
  ::在一个月里,莎莉和斯坦记录了他们手机断电多少次。 在此期间,莎莉打了55个电话,其中4个被扔了,斯坦打了36个电话,3个被扔了。 莎莉的手机断电的可能性有多大? Stan的呢?谁似乎需要更可靠的服务?

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  Sally, $\begin{align*}P(\text{dropped call})=\frac{4}{55}\approx 0.0727\end{align*}$
  ::萨利,P(窃听电话)=4550.0727

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  Stan, $\begin{align*}P(\text{dropped call})=\frac{3}{36}=\frac{1}{12}\approx 0.0833\end{align*}$
  ::Stan, P( 窃听电话) = 336= 112 0.0 833

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  Sally appears to have the more reliable service.
  ::莎莉似乎有更可靠的服务

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  Review
  ::回顾

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  Determine the following probabilities.
  ::确定下列概率。

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  In a standard deck of cards there are 4 suits (two black suits: spades and clubs, and two red suits: hearts and diamonds) and in each suit there are cards numbered 2 through 10, a jack, a queen, a king and an ace. Use this information to answer questions 1-5.
  ::在标准牌牌牌上,有4套西装(两套黑色西装:黑桃和夜总会,两套红色西装:红桃和钻石),每套西装中都有牌号为2至10、1 J、1 Q、1 王和A。使用这些信息回答问题1至5。

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  1. What is the probability of randomly drawing a queen?
     ::随机绘制女王的概率是多少?
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  2. What is the probability of randomly drawing a black card?
     ::随机绘制黑卡的概率是多少?
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  3. What is the probability of randomly drawing a face card (jack, queen or king)?
     ::随意画一张脸卡(杰克、皇后或国王)的概率是多少?
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  4. What is the probability of randomly drawing a red five?
     ::随机绘制红5的概率是多少?
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  5. What is the probability of drawing an even numbered card?
     ::绘制偶数编号卡的概率是多少?

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  Use the table of outcomes for rolling two fair dice (Example B) to answer questions 6-10.
  ::使用滚动两个公平骰子(实例B)的结果表回答问题6-10。

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  6. What is the probability of rolling a sum greater than 8?
     ::滚动金额超过8的概率是多少?
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  7. What is the probability of rolling doubles?
     ::双倍滚动的概率是多少?
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  8. What is the probability of rolling two prime numbers?
     ::滚动两个质数的概率是多少?
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  9. What is the probability of rolling a sum that is prime?
     ::滚动一个质价的概率是多少?
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  10. What is the probability of rolling an even sum?
      ::平价滚动的概率是多少?

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  In a bag of goodies at a party there are 8 gum balls, 5 gobstoppers and 10 fireball candies. When children win a party game they get to reach in the bag and pull out a prize. All three candies are spherical and the same size and thus indistinguishable to the touch ensuring a random selection.
  ::在一个聚会上,在一袋美食中,有8个口香糖球、5个口罩球和10个火球糖果。当儿童赢得了一场派对游戏后,他们就可以在袋子中伸手取出一个奖品。所有3个糖果都是球形的,大小相同,因此与触摸不相容,确保随机选择。

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  11. What is the probability of selecting a gobstopper?
      ::选择吹牛机的概率是多少?
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  12. What is the probability of selecting a fireball?
      ::选择火球的概率是多少?
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  13. You are the third person to select a candy from the bag and the first two party goers selected a gum ball and a gobstopper, respectively. What is the probability that you will get a gumball?
      ::你是从袋子中选择糖果的第三个人,而前两个派对的选手则分别选择了口香糖球和口香糖球。你能得到口香糖球的概率是多少?

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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 。