Section: 概率测量 | 12.2 概率的衡量

Section outline

Nathalie's teacher is holding a class lottery. Five winners will receive a night off from homework. There are 25 students in the class. What is the probability of Nathalie being one of the winners who will enjoy a homework-free night? Write the answer as a ratio, fraction, decimal, and percent.
::Nathalie的老师拿着一张班级彩票。 5名获奖者将享受一晚上的功课假期。班里有25名学生。 Nathalie是享受无功课之夜的优胜者之一,其概率有多大? 答案以比例、分数、小数和百分比来写。

In this concept, you will learn how to write ratios as fractions, decimals and percent.
::在此概念中,您将学习如何以分数、小数和百分比等分数、小数和百分数来记账。

Measuring Probability
::衡量概率

Ratios can be written as fractions, decimals and percents.
::比率可以以分数、小数和百分数表示。

1:2 can be written as $\begin{aligned} \frac{1}{2} \end{aligned}$ or as a decimal of .50 or as a percent of 50%
::1:2可写为12,小数点小数点或小数点为0.50,或50%的百分比。

Because probability is written as a ratio, we can also write probabilities as fractions, decimals and/or percents.
::由于概率被写成比例,我们也可以以分数、小数和(或)百分比来写概率。

Let's look at an example:
::让我们举个例子:

A bag contains 5 black ping pong balls, 8 white ping pong balls, and 7 yellow ping pong balls. What is the probability of drawing a black ping pong ball from the bag?
::包里装有5个黑乒乓球、8个白乒乓球和7个黄色乒乓球。从包里抽出一个黑乒乓球的概率是多少?

First, let’s look at writing the probability as a ratio:
::首先,让我们把概率写成比例:

\begin{aligned} P (black) & = \frac{favorable outcomes}{total outcomes} \\ favorable outcomes & = 5 \\ total outcomes & = 20 \\ P (black) & = 5 : 20 \\ = 1 : 4 \end{aligned}

::P(黑色)=有利结果合计有利结果合计=5 全部结果=20P(黑色)=5:20=1:4

Now that you have a ratio, you can easily take this ratio and write it as a fraction. Notice that the first value in the ratio becomes the numerator and the second number becomes the denominator.
::如果您有一个比率, 您可以很容易地将这个比率作为一个分数写入。请注意, 比率的第一个值变成了分子, 第二个数字变成了分母。

1:4 becomes $\begin{aligned} \frac{1}{4} \end{aligned}$
::1: 4 变成 14

Next, convert the fraction to a decimal. There are two ways to do this.
::接下来,将分数转换为小数点。有两种方法可以做到这一点。

The first way is to divide the numerator by the denominator.
::第一种方法是将分子除以分母。

$\begin{aligned} \overset{.25}{4 \bar{) 1.00}} \end{aligned}$

The decimal is .25.
::小数点后为25。

The second way is to write a proportion. You can convert one-fourth to a proportion out of 100. This only works if the denominator can be converted to 100.
::第二种方式是写一个比例。您可以将四分之一转换为 100 中的比例。只有当分母转换为 100 时才有效。

\begin{aligned} \frac{1}{4} & = \frac{◻}{100} \\ \frac{1}{4} & = \frac{25}{100} \end{aligned}

Now you can write it as a decimal: 0.25
::现在您可以把它写成小数点:0.25

Converting the decimal to a percent easy: 0.25 = 25%
::将小数点转换为简单百分比:0.25=25%

.25 or $\begin{aligned} \frac{25}{100} = 25 % \end{aligned}$
::25或25100=25%

Let’s look at another one.
::让我们再看看另一个。

Remembering the bag which contains 5 black ping pong balls, 8 white ping pong balls, and 7 yellow ping pong balls, what is the probability of choosing a yellow ping pong ball?
::记得包里有5个黑乒乓球,8个白乒乓球,7个黄乒乓球, 选择一个黄色乒乓球的概率是多少?

\begin{aligned} P (yellow) & = \frac{favorable outcomes}{total outcomes} \\ favorable outcomes & = 7 \\ total outcomes & = 20 \\ P (yellow) & = 7 : 20 \end{aligned}

::P(黄) = 有利结果合计有利结果合计= 7 全部结果= 20P(黄) = 7:20

Now, write this as a fraction.
::现在,把它写成一个分数。

$\begin{aligned} 7 : 20 = \frac{7}{20} \end{aligned}$

As a decimal:
::以小数点计:

$\begin{aligned} \frac{7}{20} = \frac{35}{100} = .35 \end{aligned}$

And as a percent:
::以百分比表示:

$\begin{aligned} .35 = 35 % \end{aligned}$

A prediction is a reasonable guess about what will happen in the future. Reasonable predictions should be based on facts. For example, you might predict that it’s going to rain today. Your prediction is reasonable if it is based on facts and evidence. For example, you might base your prediction on:
::预测是对未来会发生什么的合理猜测。合理预测应该基于事实。比如, 你可能会预测今天会下雨。如果预测是基于事实和证据,那么预测是合理的。比如, 你可能会根据以下事实和证据作出预测:

a reliable weather report you heard
::你听到可靠天气报告

dark clouds in the sky
::天空中的乌云

a satellite photo of your area
::您区域的卫星照片

a phone call from a nearby friend who lives where it’s raining
::附近朋友住在下雨的地方, 附近朋友打来电话,

The most accurate kinds of predictions are based on probability.
::最准确的预测类型是基于概率。

For example, if you assume that all 5 spinner sections in the spinner above are equal in size and, therefore, equally likely to be landed on, then the probability of spinning red is:
::例如,如果你假设以上脊柱中所有5个脊柱部分的大小相等,因此同样有可能着陆,那么旋转红色的概率是:

$\begin{aligned} P (red) = \frac{f a v o r a b l e o u t c o m e s}{t o t a l o u t c o m e s} = \frac{3}{5} \end{aligned}$
::P(red)=有利成果

Since a 3 out of 5 probability is greater than 50 percent, it is reasonable to predict that the spinner is likely to land on red.
::由于五分之三的概率超过50%,因此可以合理地预测,旋翼机很可能在红色上降落。

Here are some other reasonable predictions you might make.
::以下是你可能做出的其他合理预测。

Reasonable Prediction	Reason
The spinner is more likely to land on red than blue.	There are 3 red sections and only 1 blue section.
The spinner is equally likely to land on blue or yellow.	There spinner has an equal number of blue and yellow sections – one each.
On the average, a ratio of about 4 of 5 spins are likely turn up blue or red.	Four out of 5 sections are blue or red.
Most of the time the spinner will not land on blue or yellow.	Blue and yellow combined make up fewer than half of the sections.

Not all predictions are reasonable.
::并非所有预测都是合理的。

Unreasonable Prediction	Reason
The spinner is more likely to land on blue before it lands on red.	There are fewer blue sections (1) than red sections (3).
The spinner will never land on blue before it lands on red.	The spinner can land on blue before it lands on red; it just isn’t likely to occur.
Three out of every 5 spins will always be red.	On the average, 3 of 5 spins will be red, but on any given series of spins anything can happen.

When making predictions, it is best to keep in mind that probability predicts only what is likely to happen. All events are subject to chance. On any given event , anything can happen. If you flip a coin 4 times, you are most likely to land on heads twice and tails twice. However, that doesn’t mean that on any given pair of 4 flips, you might land on heads zero times, or you might land on heads 4 times. Neither event is likely, but both are possible and each will happen from time to time.
::在作出预测时,最好记住,概率只预测可能发生的情况。所有事件都有可能发生。在任何特定事件中,任何事情都有可能发生。如果翻硬币4次,你最可能双击头部和尾部两次。然而,这并不意味着在任何一对4次翻板上,你可能跌到头部零次,或者你可能跌到头部4次。两种情况都不可能,但两者都可能发生,而且每次都会不时发生。

The value of probability is very limited in the short term. But over the long term, predictions based on probability are usually highly accurate. In general, the greater the number of outcomes you have, the closer a prediction based on probability is likely to be.
::概率值在短期内非常有限。但从长远看,基于概率的预测通常非常准确。一般来说,结果越多,基于概率的预测就越近。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Nathalie and her teacher's lottery for a homework-less night.
::早些时候,你被问及纳塔莉和她老师的彩票的问题在一个没有家庭作业的夜晚。

Five students will be chosen from 25 total and the winners will be given a pass from homework for a night. Figure out Nathalie's probability of being chosen and write as a ratio, fraction, decimal, and percentage.
::将从25个学生中挑选5个学生,获胜者将获得一个晚上的功课通行证。找出Nathalie被选中和写作的概率,以比例、分数、小数和百分数表示。

First, identify the favorable and total outcomes :
::首先,确定有利和总的结果:

There will be 5 students who will win the lottery; so the favorable outcome is 5. There are 25 students all together; so the total outcome is 25.
::将有5名学生将中彩票;因此,有利结果为5人。共有25名学生;总结果为25人。

Next, create a ratio with equivalent fraction, decimal and percent from the result of the favorable and total outcomes. Make sure to simplify the results.

\begin{aligned} P (hard-boiled eggs) & = \frac{favorable outcomes}{total outcomes} \\ favorable outcomes & = 5 \\ total outcomes & = 25 \\ P (hard-boiled eggs) & = 5 : 25 \\ = 1 : 5 \end{aligned}

Then, create an equivalent fraction, decimal, and percentage from the ratio:
::下一步, 从有利结果和总结果的结果中, 创建一个等效分数、小数和百分数的比例。确保简化结果。 P( 硬煮蛋) = 有利结果总结果 = 5 总结果= 25P( 硬煮蛋) = 5: 25= 1: 5 然后, 创建一个等效的分数、小数和百分比 :

fraction: 1:5 = 1/5
::分数:1:5=1/5

decimal: Since the denominator can be converted to 100, either create a proportion and solve for the missing value or divide the numerator by the denominator.
::小数点:由于分母可以转换为 100, 要么为缺失值创建比例和解析, 要么将分子除以分母。

$\begin{aligned} \frac{1}{5} & = \frac{◻}{100} \\ \frac{1}{5} & = \frac{20}{100} \\ \frac{20}{100} & = 0.20 \end{aligned}$ or $\begin{aligned} \overset{.20}{5 \bar{) 1.00}} \end{aligned}$
::1510015=2010020100=0.20 或5 00=1.00 20

decimal: 0.20 = 20%
::小数点:0.20=20%

The answer is the ratio is 1:5, the fraction is 1/5, the decimal value is 0.20, and the percentage is 20%.
::答案是比例为1:5,分数为1/5,小数值为0.20,百分数为20%。

Example 2
::例2

Joan's mom hard-boiled 4 eggs to make egg salad. Joan saw the eggs on the counter and, not knowing they were hard-boiled, placed them back into the egg carton with the raw eggs. There are a dozen eggs in the carton. What is the probability that Joan's mom will choose hard-boiled eggs from the carton for her egg salad? Write your answer as a ratio, fraction, decimal, and percent.
::琼的妈妈煮了4个鸡蛋来做鸡蛋沙拉。琼在柜台上看到鸡蛋,不知道蛋是煮的,不知道蛋是煮的,就把它们放回蛋箱里,还有生鸡蛋。箱里有十几个鸡蛋。琼的妈妈在鸡蛋沙拉上从箱里选择硬煮鸡蛋的可能性有多大? 写下你的答案是比例、分数、小数和百分比。

First, identify the favorable and total outcomes:
::首先,确定有利和总的结果:

There are 4 hard-boiled eggs; so the favorable outcome is 4. There are 12 eggs all together; so the total outcome is 12.
::有4个硬锅鸡蛋, 因此有利的结果是4, 12个鸡蛋合在一起, 12个总结果就是12个。

Next, create a ratio with from the result of the favorable and total outcomes. Make sure to simplify the results.
::下一步, 从有利和总结果的结果中创建比例。确保简化结果。

\begin{aligned} P (hard-boiled eggs) & = \frac{favorable outcomes}{total outcomes} \\ favorable outcomes & = 4 \\ total outcomes & = 12 \\ P (hard-boiled eggs) & = 4 : 12 \\ = 1 : 3 \end{aligned}

::P(硬煮蛋) = 有利结果总结果 = 4 总结果 = 12P(硬煮蛋)= 4:12=1 3

Then, create an equivalent fraction, decimal, and percentage from the ratio:
::然后,从比率中得出一个相等的分数、小数小数和百分比:

fraction: 1:3 = 1/3
::分数:1:3=1/3

decimal: Since the denominator cannot be converted to 100, divide the numerator by the denominator.
::小数点:由于分母不能转换为 100, 分子除以分母。

$\begin{aligned} \overset{.33}{3 \bar{) 1.00}} \end{aligned}$ (round to the nearest hundredth)
::3,1,00,33(四舍五入至最近的第一百)

percentage: 33%
::百分比:33%

The answer is the ratio is 1:3, the fraction is 1/3, the decimal value is 0.33, and the percentage is 33%.
::答案是比例为1:3,分数为1/3,小数点为0.33,百分比为33%。

Write each ratio as a fraction, decimal and percent.
::将每种比率分别写成分数、小数和百分比。

Example 3
::例3

$\begin{aligned} 1 : 5 \end{aligned}$

First, create a fraction from the ratio:
::首先,从比例中创建一个分数 :

1:5 = 1/5

Next, create a decimal from the fraction by dividing the numerator by the denominator:
::下一步,从分数中创建小数小数,将分子除以分母 :

Since the denominator can be converted to 100, either create a proportion and solve for the missing value or divide the numerator by the denominator.
::由于分母可以转换为 100, 要么为缺失的值创建比例和解决方案, 要么将分子除以分母。

Then, convert the decimal into a percentage:
::然后,将小数点转换为百分比:

0.20 = 20%

The answer is the fraction is 1/5, the decimal value is 0.20, and the percentage is 20%.
::回答是分数是1/5,小数点值是0.20,百分比是20%。

Example 4
::例4

$\begin{aligned} 2 : 4 \end{aligned}$

First, create a fraction from the ratio, making sure to simplify if possible: 2:4 = 2/4 = 1/2
::首先,从比率中设定一个分数,确保尽可能简化:2:4=2/4=1/2

Next, create a decimal from the fraction by dividing the numerator by the denominator: Since the denominator can be converted to 100, either create a proportion and solve for the missing value or divide the numerator by the denominator.
::下一步, 从分数中创建小数点后的小数点, 将分子除以分母 : 因为分母可以转换为 100, 要么为缺失的值创建比例和解析, 要么用分母分隔数字。

$\begin{aligned} \frac{1}{2} & = \frac{◻}{100} \\ \frac{1}{2} & = \frac{50}{100} \\ \frac{50}{100} & = 0.50 \end{aligned}$ or $\begin{aligned} \overset{.50}{2 \bar{) 1.00}} \end{aligned}$
::1210012=5010050100=0.50或21.00

Then, convert the decimal into a percentage:0.50 = 50%
::然后,将小数点转换成百分比:0.50=50%

The answer is the fraction is 1/2, the decimal value is 0.50, and the percentage is 50%.
::答案是分数为 1/2,小数点值为 0.50, 百分比为 50%。

Example 5
::例5

$\begin{aligned} 4 : 5 \end{aligned}$

First, create a fraction from the ratio: 4:5 = 4/5
::首先,从比率中得出一个分数: 4: 5 = 4/5

$\begin{aligned} \frac{4}{5} & = \frac{◻}{100} \\ \frac{4}{5} & = \frac{80}{100} \\ \frac{80}{100} & = 0.80 \end{aligned}$ or $\begin{aligned} \overset{.80}{5 \bar{) 4.00}} \end{aligned}$
::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}4510045=8010080100=0.80 {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}4510045=8010080100=0.80或5 5:

Then, convert the decimal into a percentage:
::然后,将小数点转换为百分比:

0.80 = 80%

The answer is the fraction is 1/5, the decimal value is 0.20, and the percentage is 20%.
::回答是分数是1/5,小数点值是0.20,百分比是20%。

Review
::回顾

Use what you have learned about theoretical probability to answer each question.
::使用你所学到的理论概率来回答每个问题。

What is the probability of the spinner landing on 9?
::旋翼机在9号着陆的概率是多少?

List each favorable outcome.
::列出每个有利结果。

Count the number of favorable outcomes.
::计算有利结果的数目。

Count the total outcomes.
::计算总结果。

Write the probability of this event occurring.
::写入此事件的发生概率。

What is the probability of the spinner landing on 3 or 4?
::旋翼机在3或4号着陆的概率是多少?

List each favorable outcome.
::列出每个有利结果。

Count the number of favorable outcomes.
::计算有利结果的数目。

Write the probability of this event occurring.
::写入此事件的发生概率。

What is the probability of the spinner landing on blue?
::旋转器坠落蓝色的概率是多少?

List each favorable outcome.
::列出每个有利结果。

Count the number of favorable outcomes.
::计算有利结果的数目。

Write the probability of this event occurring.
::写入此事件的发生概率。

What is the probability of the spinner landing on a silver number greater than 4?
::旋转器在超过4的银号上着陆的概率是多少?

List each favorable outcome.
::列出每个有利结果。

Count the number of favorable outcomes.
::计算有利结果的数目。

Write the probability of this event occurring.
::写入此事件的发生概率。

Answer the following questions as a percent. Round your answer to the nearest tenths place.
::回答下列问题的百分率。将您回答的十进制到最近的十分位数。

A clothes dryer contains 12 socks. What is the probability of reaching inside the dryer and pulling out a blue sock?
::烘干机内有12只袜子, 挖出一只蓝袜子的可能性有多大?

What is the probability of pulling a red sock out of the dryer?
::把红袜子从烘干机里拉出来的可能性有多大?

What is the probability of pulling a blue or white sock out of the dryer?
::从烘干机中拉出蓝色或白色袜子的概率是多少?

What is the probability of pulling a blue or red sock out of the dryer?
::从烘干机中拉出蓝色或红色袜子的概率是多少?

Review (Answers)
::回顾(答复)

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。

Section outline

Measuring Probability ::衡量概率

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)