经验概率

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Empirical Probability 
::经验概率

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A probability simulation is an experiment designed to determine a probability from many trials. By looking at the number of favorable outcomes and dividing by the number of trials, we can get an estimate of the true probability. The more trials we can do, the better our estimate of the true probability will be, but given that we can’t do infinitely many trials, the result we get will always be just an estimate of the true probability. Often we perform probability simulations because we can’t determine theoretical probability from looking at the sample space .
::概率模拟是一种旨在确定从许多试验中得出的概率的实验。 通过查看有利结果的数量并除以试验数量,我们可以对真实概率做出估计。 我们所能做的试验越多,对真实概率的估算就越好,但鉴于我们无法做无限多的试验,我们所得到的结果总是只是对真实概率的估算。 我们进行概率模拟往往因为我们无法确定从抽样空间看的理论概率。

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Real-World Application: TV Repair 
::真实世界应用程序:电视修理

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A cable TV company sends out a repair technician to replace faulty receiver boxes. The company has boxes made by Panasonic and Scientific Atlanta , both in equal quantities. The technician carries 3 of each type of box in his van, and always replaces a box with one of the same brand. If the technician visits 4 homes before returning back to the depot, determine the probability that he will not have enough of one box type to make all the needed replacements.
::一家有线电视公司派出一名修理技师来更换有缺陷的接收箱。 该公司拥有由Panasonic和Science Atlanta公司以同等数量制成的箱子。 技师在他的面包车内携带每类箱子中的3个,并且总是用同一个品牌来替换一个箱子。 如果技师在返回仓库前访问4个家庭,那么他是否有足够的一个箱子类型来替换所有需要的箱子。

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This is a situation that we can model far more easily than we could conduct a real-life experiment. Since there are equal numbers of both boxes, we need to set up a model with a probability of $\begin{array}{r}\frac{1}{2}\end{array}$ for each element, like a coin toss. Visiting four houses where each house has an equal chance of needing one type of box or the other is like flipping a coin four times. Getting four heads or four tails is like needing four of one type of box, which is the only situation where the technician would not have enough of one type.
::这是一种我们比实际实验更容易模拟的情况。 由于两个盒子的数量相同,我们需要为每个元素建立一个12个概率的模型,比如抛硬币。 访问四所房屋,每所房屋同样需要一种或另一种类型的盒子,就像翻硬币四次。 获得四头或四种尾巴就像需要一种盒子中的四头或四种,这是技术员唯一没有一种类型足够能力的情况。

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So let’s suppose we flip four coins 50 times and record the results, and they look like this:
::让我们假设我们翻了四枚硬币五十倍, 记录结果,

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Out of 50 trials, it’s as if the technician required four of the same type of box 6 times. So we can say that the probability that the technician will run out of one box type is approximately $\begin{array}{r}\frac{3}{25}\end{array}$ or 0.12 .
::50次试验中,技术员需要4次相同的箱6次。 因此我们可以说,技术员用完一个箱的概率大约是325或0.12。

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Solving Using Theoretical Probability
::利用理论概率解决

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The problem in the first example can also be solved by finding the theoretical probability. Let's see how to do that.
::第一个例子中的问题也可以通过找到理论概率来解决。让我们来看看如何做到这一点。

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Notice that instead of actually flipping the coins a lot of times, we could have used our earlier knowledge of theoretical probability and the sample space for four coin flips:
::我们本可以利用我们早期的理论概率知识 和四个硬币的样本空间:

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Here, we can see that we would expect the technician to run out of one type of box about 2 times out of 16, so the probability is about $\begin{array}{r}\frac{1}{8}\end{array}$ or 0.125 . But if we hadn’t known for sure what the odds of getting heads or tails on each flip were, we wouldn’t have been able to calculate the odds of getting four heads or tails this way, so we would have had to find out by experiment instead.
::在这里,我们可以看到,我们期望技术员在16个箱子中能用完大约2次,因此概率大约是18或0.125。 但如果我们不确定每只翻转机头或尾巴的概率,我们就无法计算出这样得到4个头或尾的概率,因此我们不得不通过实验来找出答案。

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We also can check probability calculations like this against actual experimental data to see for ourselves whether something happens as often as we’d expect it to. If it doesn’t, something might be going on that we need to investigate.
::我们还可以对照实际实验数据来检查这样的概率计算,看看是否像我们预期的那样经常发生一些事情。 如果不发生,我们也许需要调查一些事情。

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Finding the Experimental Probability of an Event
::查找事件实验概率

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In the previous examples, we saw how theoretical probability can be approximated by doing an experiment. We used a results table to approximate the probability of a certain event occurring (the technician running out of either type of box). We can approximate the probability of an event by using:
::在前几个例子中,我们看到了如何通过进行实验来估计理论概率。我们用一个结果表来估计发生某一事件的概率(技师已经用完任何一种方框)。我们可以使用下列方法来估计事件概率:

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::P(E)- 匹配活动数目 审判总数

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Randomness in the results will mean that we always get an approximation of the true probability, but the more trials we do, the more accurately our experimental probability will match the theoretical probability.
::结果的随机性将意味着我们总是得到 真实概率的近似值, 但是越多的试验, 我们的实验概率就越准确 将符合理论概率。

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Nadia and Peter are playing dice, but Peter keeps winning and Nadia suspects him of cheating. She is suspicious about the number of times Peter rolls a six, and so she conducts the following experiment: She rolls the suspect die 100 times, writing down the result each time. The results are:
::Nadia和Peter在玩骰子,但Peter一直赢,Nadia怀疑他作弊。她怀疑Peter滚动六次的次数,因此她进行了以下实验:她把嫌疑人的死100次,每次写下结果。结果如下:

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Organize the data in a table and determine if 6 is more likely to come up then the other numbers.
::将数据组织在一个表格中,确定6个数据是否更有可能出现,然后确定其他数字。

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Here’s what we get if we tally up all the results in a table:
::如果我们在表格中统计所有结果,

Number	Tally	Total	P( number )
1	$$\begin{array}{r}\bcancel{||||}\bcancel{||||}\bcancel{||||}\end{array}$$	15	$$\begin{array}{r}P(1)=\frac{15}{100}\approx 0.15\end{array}$$
2	$$\begin{array}{r}\bcancel{||||}\bcancel{||||}||||\end{array}$$	14	$$\begin{array}{r}P(2)=\frac{14}{100}\approx 0.14\end{array}$$
3	$$\begin{array}{r}\bcancel{||||}\bcancel{||||}\bcancel{||||}\end{array}$$	15	$$\begin{array}{r}P(3)=\frac{15}{100}\approx 0.15\end{array}$$
4	$$\begin{array}{r}\bcancel{||||}\bcancel{||||}|||\end{array}$$	13	$$\begin{array}{r}P(4)=\frac{13}{100}\approx 0.13\end{array}$$
5	$$\begin{array}{r}\bcancel{||||}||||\end{array}$$	9	$$\begin{array}{r}P(5)=\frac{9}{100}\approx 0.09\end{array}$$
6	$$\begin{array}{r}\bcancel{||||}\bcancel{||||}\bcancel{||||}\bcancel{||||}\bcancel{||||}\bcancel{||||}||||\end{array}$$	34	$$\begin{array}{r}P(6)=\frac{34}{100}\approx 0.34\end{array}$$

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It’s clear looking at the table that something strange is going on with the die in question – 6 occurs approximately twice as often as the other numbers, so we could reasonably assume that the die is weighted unfairly. However, we still can’t be 100% certain that the results we are seeing are not just due to chance. We must therefore talk only in terms of likelihood , and not certainty .
::显而易见的是,从表上看,有关死亡正在发生一些奇怪的事情 — — 6起大约是其他数字的两倍,因此我们可以合理地认为死亡是不公平的。 然而,我们仍然不能百分之百地确定我们所看到的结果不仅仅是偶然的。 因此,我们必须只谈可能性,而不是确定性。

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Example
::示例示例示例示例

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Example 1
::例1

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Juan suspects that his lucky coin is actually weighted so that he gets heads more often than tails in a coin toss. The reason he is suspicious is because he seems to get several heads in a row when he tosses it. He conducts a probability simulation and gets the following results:
::Juan怀疑他幸运的硬币其实是加权的,这样他比抛硬币的尾巴更经常地得到头。 他怀疑的原因是,当他抛硬币时,他似乎会连续得到几个头。 他进行概率模拟,并获得以下结果:

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$\begin{array}{r}HTTHHHHTHHTTTHHHHHH\end{array}$
::呼呼呼呼呼呼呼呼的呼呼呼

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$\begin{array}{r}HHTTHTHHTHTTTHHTHTTHT\end{array}$
::希图图图图图图图

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What is the experimental probability of getting heads with Juan's coin in this case?
::在这个案子中,用Juan的硬币 获得头的实验概率是多少?

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First, find the number of total tosses, the number of heads, and the number of tails:
::首先, 找到所有切片的总数、 头数和尾部数 :

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40 tosses
::40 个推进点

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23 heads
::23个首长23个首长

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17 tails
::17尾

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This means that the experimental probability of getting heads with Juan's coin is:
::这意味着,用胡安的硬币来获取头的实验概率是:

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::P(E)-匹配活动的数量 审判总数=2640=0.65

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For the coin to be fair, we would expect heads around 50% of the time. 65% is higher than 50%, but this difference may be due to random chance. Further experimentation would help to get a more accurate answer.
::为使硬币公平,我们预计大约50%的时间会到来。 65%比50%高,但这一差异可能是随机的。 进一步的实验将有助于获得更准确的答案。

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Review
::回顾

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1. Peter and Andrew each visit the hardware store in the high street every week. The store is open 6 days a week (it is closed on Sundays) and Peter and Andrew visit the store on random days when it is open.
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   1. Use a pair of dice to simulate what day Andrew and Peter each visit the store, and determine experimentally the probability that they both visit the store on the same day.
      ::用一对骰子来模拟安德鲁和彼得 每一天去商店的那一天 并实验地确定他们在同一天 都去商店的概率
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   2. What would you expect the theoretical probability to be?
      ::你会期望理论概率是什么呢?
   
   ::Peter 和 Andrew 每周都去高街的五金店。 商店每周营业6天( 星期日不营业 ) , Peter 和 Andrew 随机到商店营业。 使用一对骰子模拟Andrew 和 Peter 每人去商店的那天, 并实验性地确定他们在同一天都去商店的概率。 您对理论概率有什么期望呢 ?
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2. Find experimentally both the probability and odds for the next car passing a stoplight being red if the previous 25 car colors were: red, blue, white, blue, silver, red, black, black, white, red, green, red, black, blue, white, red, silver, white, red, black, white, blue, silver, red, black.
   ::如果前25辆车的颜色是:红色、蓝色、白色、蓝色、蓝色、银色、红色、黑色、黑色、黑色、白色、红色、红色、红色、黑色、黑色、黑色、红色、黑色、蓝色、白色、白色、红色、银色、白色、红色、蓝色、红色、蓝色、红色、白色、白色、蓝色、银色、银色、红色、红色、黑色,

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For 3-13, determine whether you could calculate the theoretical probability of the given event based on your knowledge of the possible outcomes, or whether you would have to do a test (or get more real-world information some other way) to find the experimental probability:
::对于 3-13, 确定您是否可以根据您对可能结果的了解来计算给定事件的理论概率, 或者您是否必须做一个测试( 或以其他方法获得更多真实世界信息) 才能找到实验概率 :

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3. Flipping a coin three times in a row and getting three heads.
   ::连续三次抛掷一枚硬币 并获得三个头。
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4. Pulling a nickel from your pocket when you know you have three nickels and five dimes in your pocket.
   ::当你知道你口袋里有三分五分钱时 就从口袋里掏出一分钱
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5. Pulling a nickel from your pocket when you know you have ten coins in your pocket but can’t remember what they are.
   ::当你知道你口袋里有十块硬币,
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6. Guessing the right answer on a multiple-choice question.
   ::猜对一个多选题的正确答案
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7. Guessing the right answer on a free-response question.
   ::猜测一个自由回答问题的正确答案
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8. Getting a perfect score on a twenty-question multiple-choice test.
   ::获得一个完美的分数 在20个问题 多重选择测试。
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9. Getting a perfect score on a test that has ten multiple-choice questions and ten free-response questions.
   ::测试有一个完美的得分 测试有10个多重选择问题 10个自由回答问题
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10. Guessing a randomly chosen high school student’s age correctly.
    ::随机选择的高中学生年龄猜对了。
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11. Sharing a birthday with one of your three best friends.
    ::跟你三个最好的朋友之一 一起过生日
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12. Getting a flat tire while driving home.
    ::开车回家的时候 车胎爆了
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13. Having your left front tire be the one that goes flat, whenever you do get your next flat tire.
    ::你左前胎是那个爆胎的, 当你得到你的下一个爆胎时。

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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 。