节: 基本概率 - 概率和统计 | 6.1 基本概率 - 概率和统计

章节大纲

Most people have heard, I think, of the old adage that buttered bread always lands buttered side down. However, from a scientific standpoint, what is the real statistical and experimental probability of buttered bread landing butter side up? For that matter, what is the difference between a statistical and an experimental probability?
::我认为,大多数人都听过黄油面包的老格言,黄油面包总是会掉在黄油一边。然而,从科学的角度来看,黄油面包着陆黄油的统计和实验可能性究竟有多大? 就此而言,统计和实验概率有什么区别?

Basic Probability
::基本概率

Probability is the study of chance. When studying probability, there are two very general classifications: theoretical probability and experimental probability .
::概率是概率研究。在研究概率时,有两个非常一般的分类:理论概率和实验概率。

Theoretical probability is the calculated probability that a given outcome will occur if the same experiment were completed an infinite number of times.
::理论概率是,如果同一实验完成无限次数,计算得出某一结果的概率。

Experimental probability is the observed result of an experiment conducted a limited number of times.
::实验概率是所观测到的实验结果,该实验进行了有限的几次。

For example, ignoring the very slight differences between the figures stamped onto each side of a coin, the statistical probability of a coin landing heads-up is 50%. However, if you flip a coin 10 times, you may very well find that the observed experimental probability results in 60% or 70% or even greater probability of one side landing up. This discrepancy is perfectly natural and expected when conducting experiments, and it is important to recognize it.
::例如,忽略了硬币每面所盖数字之间的微小差异,硬币着陆头顶的统计概率为50 % 。但是,如果翻硬币10次,你可能会发现观察到的实验概率导致一方着陆60%或70%,甚至更大。在进行实验时,这种差异是完全自然的和预期的,而且必须认识到这一点。

In this lesson we will confine our study to the probability of a simple event . The probability of a simple event is the calculated chance of a specific direct outcome of a single experiment where in all possible outcomes are equally likely. To calculate the probability of such an outcome, we use a very simple and intuitive formula:
::在这个教训中,我们的研究将限于一个简单事件的概率。一个简单事件的概率是计算出一个单一实验的具体直接结果的概率,在所有可能的结果都同样可能。为了计算这种结果的概率,我们使用一个非常简单和直观的公式:

$\begin{aligned} P (x) = \frac{number of events where x is true}{total number of possible events} \end{aligned}$
$\begin{align*}P(x)=\frac{\text{number of events where} \ x \ \text{is true}}{\text{total number of possible events}}\end{align*}$
::P(x) = x 是真实的事件数量, 可能的事件总数

Where “ $\begin{align*}P(x)\end{align*}$ ” is the probability that $\begin{align*}x\end{align*}$ will occur
::“P(x)”是指X的概率的“P(x)”

In other words, just as you might expect, the probability of randomly picking one of the three blue marbles out of a bag with ten marbles total would be $\begin{align*}\frac{3}{10}\end{align*}$ .
::换句话说,正如你可能预期的那样,从一个袋子中随机取出三个蓝色弹珠之一,总共十个弹珠的概率是310。

Calculating Probability
::计算概率

1. You are given a big containing 15 equally sized marbles. You know there are 10 yellow marbles and 5 green marbles in the bag. What is the probability that you would pull a yellow marble out, if you reach in the bag and grab a marble at random?
::1. 给您一个大块,内装15个相同大小的大理石;你知道包里有10个黄色大理石和5个绿色大理石;如果你在袋子里随意拿大理石,你拔出黄色大理石的可能性有多大?

Use the formula for the probability of a simple event:
::使用简单事件概率的公式 :

$\begin{aligned} P (x) = \frac{number of outcomes where x is true}{total number of possible outcomes} \end{aligned}$
$\begin{align*}P(x)=\frac{\text{number of outcomes where} \ x \ \text{is true}}{\text{total number of possible outcomes}}\end{align*}$
::P(x)=以x为真实结果的结果数,而x为可能结果总数

In this case, we have:
::在这种情况下,我们有:

$\begin{aligned} P (y e l l o w) = \frac{10 y e l l o w m a r b l e s}{15 t o t a l m a r b l e s} \end{aligned}$
$\begin{align*}P(yellow)=\frac{10 \ yellow \ marbles}{15 \ total \ marbles}\end{align*}$
::P(黄)=10黄圆珠15

Which would reduce to:
::这将减少:

$\begin{aligned} P (y e l l o w) = \frac{2}{3} o r 66.6 \bar{6} % \end{aligned}$
$\begin{align*}P(yellow)=\frac{2}{3} \ or \ 66.6 \bar{6} \% \end{align*}$
::P(黄)=23或66.66

2. What is the probability of rolling an odd number on a standard six-sided die?
::2. 在标准的六方死亡上滚动奇数的可能性有多大?

A standard die has three odd numbers (1, 3, 5) and three even numbers (2, 4, 6). Therefore, the probability of rolling an odd number is:
::标准死亡有三个奇数(1、3、5)和三个偶数(2、4、6)、三个偶数(2、4、6)。因此,一个奇数滚动的概率是:

$\begin{aligned} P (o d d) = \frac{3 o d d}{6 t o t a l} \end{aligned}$
$\begin{align*}P(odd)=\frac{3 \ odd}{6 \ total}\end{align*}$
::P(od)=3奇特6总计

Reducing to:
::减少:

$\begin{aligned} P (o d d) = \frac{1}{2} o r 50 % \end{aligned}$
$\begin{align*}P(odd)=\frac{1}{2} \ or \ 50 \%\end{align*}$
::P(odd)=12或50%

Finding Experimental Probability
::实验概率

If Lawrence is playing with a standard 52-card deck, then the statistical probability of him pulling a single Queen at random is: $\begin{align*}\frac{4 \ queens}{52 \ cards}=\frac{1}{13}=7.7 \%\end{align*}$ . If he decides to test it out and ends up pulling a Queen at random 6 times in 52 trials of “pull a card, record it, put it back”, what is the experimental probability of pulling a Queen?
::如果劳伦斯在玩标准的52张牌牌牌牌,那么他随机拉一个女王的统计概率是:4 Q52卡=113=7.7%。如果他决定测试它,并在52次“拉一张卡片,记录下来,放回去”试验中以6次随机拉一个女王,那么拉一个女王的实验概率是什么?

Recall that experimental probability is the observed probability of a number of identical experiments. Experimental probability is not affected by statistical probability (it may be predicted by it, but not affected ), therefore the experimental probability is:
::回顾实验概率是若干相同实验的观测概率。实验概率不受统计概率的影响(它可能预测,但不会受到影响),因此实验概率如下:

%3D%5Cfrac%7B6%20%5C%20Queens%7D%7B52%20%5C%20trials%7D">
$\begin{aligned} P (y) = \frac{6 Q u e e n s}{52 t r i a l s} \end{aligned}$
$\begin{align*}P(y)=\frac{6 \ Queens}{52 \ trials}\end{align*}$
::P=6 Queens52审判

Reducing to:
::减少:

%3D%5Cfrac%7B3%7D%7B26%7D%3D11.5%20%5C%25">
$\begin{aligned} P (y) = \frac{3}{26} = 11.5 % \end{aligned}$
$\begin{align*}P(y)=\frac{3}{26}=11.5 \%\end{align*}$
::P=326=11.5%

Earlier Problem Revisited
::重审先前的问题

From a scientific standpoint, what is the real statistical and experimental probability of buttered bread landing butter side up? For that matter, what is the difference between a statistical and an experimental probability?
::从科学角度看,黄油面包上岸黄油的实际统计和实验概率是多少? 就此而言,统计和实验概率有什么区别?

Remember that the difference is that statistical probability is the calculated probability of a specific outcome, and experimental probability is the observed probability.
::记住,差别在于统计概率是计算出某一结果的概率,实验概率是观察到的概率。

The statistical probability of the bread landing butter side up can be assumed to be $\begin{align*}\frac{1}{2}\end{align*}$ , based on bread having two sides.
::根据面包有两面,可以假定面包上岸黄油的统计概率为12。

According to the “MythBusters” experiment in the video, the observed probability was $\begin{align*}\frac{29}{45}\end{align*}$ . However, you should know that your results might be different!
::根据视频中的“神话信使”实验,观察到的概率是2945。但是,你应该知道,你的结果可能不同!

Examples
::实例

Example 1
::例1

What is the probability of pulling the 1 red marble out of a bag with 12 marbles in it?
::用12颗大理石从袋子里拉出1个红色大理石的概率是多少?

$\begin{align*}P(red)=\frac{\text{1 red marble}}{\text{12 total marbles}}=\frac{1}{12} \ or \ 8.3 \%\end{align*}$
::P(red)=1红色大理石12

Example 2
::例2

What is the probability of a spinner landing on “6” if there are 6 equally spaced points on the spinner?
::如果在“6”上有6个相同的空格点,旋转器降落在“6”上的概率是多少?

$\begin{align*}P(red)=\frac{\text{1 red marble}}{\text{12 total marbles}}=\frac{1}{12} \ or \ 8.3 \%\end{align*}$
::P(red)=1红色大理石12

Example 3
::例3

What is the probability of pulling a red card at random from a standard deck?
::从标准甲板随机拉红色卡的概率是多少?

$\begin{align*}P(red)=\frac{\text{26 red cards}}{\text{52 total cards}}=\frac{26}{52}=\frac{1}{2} \ or \ 50 \%\end{align*}$
::P(red)=26 红卡52共卡片=2652=12=12或50%

Example 4
::例4

What is the experimental probability of heads in an experiment where Scott flipped a coin 50 times and got heads 21 times?
::史考特把硬币翻了50次头又翻了21次实验中头头的实验概率是多少?

$\begin{align*}P(heads)=\frac{\text{21 heads}}{\text{50 flips}}=\frac{21}{50} \ or \ 42 \%\end{align*}$
::P(头) = 21头 = 21头 = 250 头翻 = 2150 或 42%

Example 5
::例5

What is the probability of shaking the hand of a female student if you randomly shake the hand of one person in a room with 23 female students and 34 male students?
::与23名女学生和34名男学生在一间房间里随机握着一个人的手,女生握手的机率是多少?

$\begin{align*}P(female)=\frac{\text{23 females}}{\text{57 students }}=\frac{23}{57} \ or \ 40.4 \%\end{align*}$
::P(女性)=23 女生57人=2357人或40.4%

Review
::回顾

Questions 1-10, find the probability:
::问题1-10,发现概率:

Rolling a 4 on a standard die
::以标准死亡为标准死亡标准滚4

Pulling a King from a standard deck
::从标准甲板上拉国王

Pulling a green candy from an opaque bag with 5 red, 3 yellow, 3 blue, and 6 green candies.
::从一个不透明的袋子里拿出绿色糖果 5个红色 3个黄色 3个蓝色 6个绿色糖果

Getting a 5 from one spin on a spinner numbered 1-8 (equally spaced)
::在一个编号 1-8 的旋转器上从一个旋转得到 5 个( 平均间距)

Rolling an even number on a 20-sided die
::20度死亡时偶数滚动

Rolling and odd number on a standard die
::标准死亡的滚动和奇数

Pulling a red card from a standard deck
::从标准甲板上拉红牌

Pulling a face card from a standard deck
::从标准甲板上拉一张脸牌

Spinning red on a spinner with Red, Orange, Yellow, Green, Blue and Purple (equally spaced)
::红色、橙色、黄色、绿色、绿色、蓝色和紫色(平间距)

Pulling a club from a standard deck
::从标准甲板上拉一个俱乐部

Pulling a brown candy from a box of 25 candies, containing equal numbers of brown, red, green, blue, and yellow candies
::从25个糖果盒里拿出一盒棕色糖果里面有相同数量的棕色、红色、绿色、绿色、蓝色和黄色糖果

Getting a prime number with a random number generator that has an equal chance of generating any number between 1 and 50
::获得一个有随机数生成器的质号,该源具有生成任何1至50之间数字的同等机会

Getting a composite number with the same generator
::获得同一个发电机的复合数字

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。

基本概率 - 概率和统计

章节大纲

Basic Probability
::基本概率

Calculating Probability
::计算概率

%3D%5Cfrac%7B3%7D%7B26%7D%3D11.5%20%5C%25">
$\begin{aligned} P (y) = \frac{3}{26} = 11.5 % \end{aligned}$
$\begin{align}P(y)=\frac{3}{26}=11.5 \%\end{align}$
::P=326=11.5%

Earlier Problem Revisited
::重审先前的问题

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)

章节大纲

Basic Probability ::基本概率

Calculating Probability ::计算概率

%3D%5Cfrac%7B3%7D%7B26%7D%3D11.5%20%5C%25"> P ( y ) = 3 26 = 11.5 % \begin{align*}P(y)=\frac{3}{26}=11.5 \%\end{align*} ::P=326=11.5%

Earlier Problem Revisited ::重审先前的问题

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)

Basic Probability
::基本概率

Calculating Probability
::计算概率

%3D%5Cfrac%7B3%7D%7B26%7D%3D11.5%20%5C%25">
$\begin{aligned} P (y) = \frac{3}{26} = 11.5 % \end{aligned}$
$\begin{align}P(y)=\frac{3}{26}=11.5 \%\end{align}$
::P=326=11.5%

Earlier Problem Revisited
::重审先前的问题

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)