Section: 分析公平和裁决的概率 | 11.9 分析公平和裁决的概率-interactive

Section outline

The word "fair" is frequently used informally, often having to do with the inequities that exist in our world:
::“公平”一词经常被非正式地使用,常常与我们世界上存在的不平等有关:

"It's not fair that Mark has more video games than me."
::"这不公平,马克有更多的电子游戏比我多。"

"It's not fair that some people don't have enough money for food."
::"有些人没有足够的钱去吃东西,这不公平"

"It's not fair that I have to take a Spanish class."
::"我必须上西班牙语课是不公平的"

Fairness in the above situations can be a matter of opinion.
::在上述情况下,公平问题可能是一个意见问题。

The word "fair" is also used formally, in the context of games and making choices. A basic game of chance is considered fair if every player has an equal probability of winning. A choice is fair if all possible options have an equal probability of being chosen. You can use your knowledge of probability to analyze the fairness of games and make fair choices.
::“ 公平” 这个词在游戏和做出选择时也正式使用。如果每个玩家都有同等的胜选概率, 基本的机会游戏就被认为是公平的。如果所有可能的选项都有同等的被选中概率, 选择也是公平的。您可以使用您对概率的知识来分析游戏的公平性和做出公平的选择。

Fairness
::公平公正

A bag contains 10 red marbles and 1 silver marble. You and your brother decide that you will reach into the bag and select a marble. If you choose a red marble then you have to do the dishes and if you choose a silver marble then he has to do the dishes. Is this fair? Describe a fair way to choose who does the dishes by choosing marbles out of a bag. Explain why this is fair using probabilities.
::袋子里装有10个红色大理石和1个银银大理石。你和你哥哥决定要伸进袋子里挑选大理石。如果你选择了红色大理石,那你就得洗碗。如果你选择了银大理石,他必须洗碗。这样公平吗?用袋子里选择大理石来描述一个选择谁洗碗的公平方式;用概率解释为什么这是公平的。

To be fair, both you and your brother should have an equal chance of having to do the dishes. This means the probability that you will do the dishes should be $\begin{aligned} \frac{1}{2} (50 %) \end{aligned}$ and the probability that your brother will do the dishes should be $\begin{aligned} \frac{1}{2} (50 %) . \end{aligned}$
::公平地说,你和你的兄弟都应该有同样的机会做盘子。这意味着你洗盘子的概率应该是12(50%),你弟弟洗盘子的概率应该是12(50%)。

You will have to do the dishes if you choose a red marble. There are 10 red marbles and 11 marbles total. The probability of you doing the dishes is $\begin{aligned} \frac{10}{11} \approx 91 % . \end{aligned}$
::如果您选择了红色大理石, 您将不得不洗盘子。共有 10 个红大理石和 11 个大理石。您洗盘子的概率是 1011 91% 。

Your brother will have to do the dishes if you choose the silver marble. There is 1 silver marble and 11 marbles total. The probability of your brother doing the dishes is $\begin{aligned} \frac{1}{11} \approx 9 % . \end{aligned}$
::如果你选择银大理石,你哥哥就得洗盘子。有1个银大理石和11个大理石。你哥哥洗盘子的概率是111.9%。

This is definitely not a fair way to decide who will do the dishes, because you are much more likely to end up doing the dishes.
::这绝对不是决定谁洗盘子的公平方法,因为你更有可能最终洗盘子。

Probability
::概率

1. Your math class has 32 students. Your teacher, Ms. Peters, needs to choose 5 people randomly to make a presentation. How can she use her calculator to help her choose 5 people at random?
::1. 你的数学班有32名学生,你的老师Peters女士需要随机挑选5人来做演讲,她如何使用计算器帮助她随机选择5人?

Random in this sense means that Ms. Peters wants each student to have an equal chance of being chosen. If she just chooses 5 people and tries to be random, her choices likely won't actually be random. She might choose people who haven't presented recently, or try to get a mix of boys and girls.
::从这个意义上说,随机决定意味着Peters女士希望每个学生都有被选中的同等机会。如果她只选择了5个人,并试图随机选择,那么她的选择可能不会是随机的。她可能选择最近没有出现的人,或者尝试让男孩和女孩混在一起。

To make a truly random choice, it helps to use a random number generator. First, she should assign each person in the class a number from 1 to 35 (this could be done in alphabetical order or in any other way). Then, she should use the randInt( function on her graphing calculator. This function produces a given number of random integers between two integers. It is found on a TI-83/ TI-84 by pushing the MATH button, then scrolling right to PRB , then scrolling down to 5: randInt( . Back on the home screen, type the lower limit of 1, the upper limit of 35, and 5 (for the 5 numbers that you want produced). Press ENTER to produce 5 random numbers between 1 and 35.
::为了做出真正随机的选择, 它会帮助使用随机数字生成器。首先, 她应该指定班级中每个人的编号从 1 到 35 个( 这可以按字母顺序或以任何其他方式进行 ) 。然后, 她应该使用 RandInt ( 图形计算器上的函数 ) 。此函数在两个整数之间产生一定数量的随机整数。在 TI- 83/ TI- 84 上, 按下 MATH 按钮, 然后向 PRB 滚动, 然后向 PRB 滚动到 5 个 : RandInt (. 回到主屏幕上, 键入 1 的下限, 上限为 35 和 5 。按 ENTER 键在 1 到 35 之间生成 5 个随机数。

This time, students 34, 32, 6, 19, and 15 should make the presentation. Note that it is possible that the same number could have been chosen twice. If that happened, Ms. Peters could just randomly generate another number.
::这一次,学生34、32、6、19和15应作介绍。请注意,同一数字有可能被选中两次。如果发生这种情况,Peters女士可以随机生成另一个数字。

Graphing Calculator Randomness: Even the graphing calculator isn't truly random. It is following a formula to create numbers that appear random. Using their default setting, all calculators will produce the same random numbers in the same order, because they are using the same formula. If you wish to change the starting number your calculator is using in its formula to create random numbers, type in a number (any number), then press STO> , then use the rand function and press ENTER . Now the random numbers your calculator generates will be different from the random numbers generated by another calculator.
::图形计算随机性 : 即使图形计算计算器不是真正随机的。它也遵循一个公式来创建随机的数字。使用默认设置, 所有计算器都会以相同的顺序产生相同的随机数字, 因为他们使用的是相同的公式。如果您想要改变您的计算器在其公式中使用的起始数字, 以创建随机数字, 键入一个数字( 任意数字) , 然后按下STO> , 然后使用兰特函数, 按下 ENTER 。现在, 您计算器生成的随机数字将不同于另一个计算器生成的随机数字。

2. Your history class taught by Mr. Bliss has 35 students. On the first day of school Mr. Bliss asks everyone when their birthday is. You are surprised to learn that two people in your class have the same birthday. You say "I bet if we looked at the birthdays for each history class taught by Mr. Bliss, none of the other classes will have a birthday shared by at least two people." If Mr. Bliss teaches five history classes, each with 35 students, is this a good bet?
::2. Bliss先生教授的历史课有35名学生,Bliss先生在开学第一天问每个人他们的生日是什么,Bliss先生问每个人。你很惊讶地知道你班里有两个人生日是同一个人。你说,“如果我们看看Bliss先生教授的历史课的生日,那么其他班里的任何一个生日都不会至少有两个人共享。 ”如果Bliss先生教五个历史课,每个班有35名学生,这是个好赌吗?

Consider the probability of a group of 35 people NOT having the same birthday. In order to find this probability you need:
::考虑35人组没有同一生日的概率。为了找到这一概率,您需要 :

The number of ways to assign different birthdays (from the 365 days in a year) to 35 people.
::给35人分配不同生日(每年365天)的方法数量。

The number of ways to assign a birthday (from the 365 days in a year) to each of 35 people (repeats okay).
::给35人每人分配生日(从每年365天算起)的方法数量(可重写)。

You can use the for these calculations
::您可以使用这些计算

1. To assign a different day to each person, there are 365 choices for person #1, 364 choices for person #2, 363 choices for person #3, and so on.
::1. 为给每个人分配不同的一天,1、364人有365个选择,2、363人有363个选择,3等。

Number of ways to have 35 distinct birthdays: $\begin{aligned} 365 \cdot 364 \cdot 363 \dots 333 \cdot 332 \cdot 331 \end{aligned}$
::拥有35个不同生日的方法数量:365364363}3333}332}331

2. To assign a day to each person where repeats are okay, there are 365 choices for person #1, 365 choices for person #2, and so on.
::2. 给每个人分配一天,如果重复情况正常,那么第1号、第2号、第365号等的人就有365个选择。

Number of ways to pick birthdays for 35 people: $\begin{aligned} 365 \cdot 365 \dots 365 \cdot 365 = 365^{35} \end{aligned}$
::35人选择生日的方法数量:365 365 365 365 365 365 365365 365365=365 355

The probability of 35 distinct birthdays can be calculated with the help of a computer to be:
::在计算机的帮助下,35个不同的生日的概率可以计算为:

$\begin{aligned} \frac{365 \cdot 364 \cdot 363 \dots 333 \cdot 332 \cdot 331}{365^{35}} \approx 18.6 % \end{aligned}$

If the probability of everyone have a different birthday is 18.6%, then the probability of at least two people sharing a birthday (which is a complementary probability), is $\begin{aligned} 100 % - 18.6 % = 81.4 % . \end{aligned}$ Even though it might seem rare, there is actually a pretty high probability that in a group of 35 people there will be a shared birthday.
::如果每个人不同生日的概率是18.6%,那么至少两个人共同过生日(这是补充性概率)的概率是10018.681.4 % 。尽管这似乎很少见,但事实上在35人中,有一个共同生日的概率是相当高的。

If Mr. Bliss teaches 5 history classes, it's definitely possible that more than one class will have a shared birthday within it. Therefore, you did not make a good bet.
::如果Bliss先生教5个历史课, 肯定不止一个课里会有一个共同的生日。因此,你没有下好的赌注。

This is a classic probability problem with a result that surprises most people!
::这是一个典型的概率问题其结果让大多数人惊讶!

Examples
::实例实例实例实例

Example 1
::例1

Your friend Jeff is on a game show hoping to win a car. The host of the game show reveals three doors, and tells Jeff that the car is behind one of them. Behind the other two doors are goats. The game show host knows where the car is.
::你的朋友杰夫在一场比赛中希望赢得一辆车。游戏主持人展示了三扇门,告诉杰夫车在其中一扇门后面。另外两扇门后面是山羊。游戏主持人知道车在哪里。

The game show host tells Jeff to pick a door and he does. Then, the game show host opens one of the doors that Jeff did NOT choose to reveal a goat.
::游戏节目主持人让杰夫挑扇门,他就这样做了。然后,游戏节目主持人打开了杰夫没有选择暴露羊的大门之一。

The game show host asks Jeff if he wants to switch doors. Jeff says yes, because he believes that his chance of winning is greater if he switches his choice.
::游戏节目主持人问杰夫他是否想打开门。杰夫说是的,因为他相信,如果他改变选择,他赢的机会更大。

Did Jeff make the right decision?
::杰夫做了正确的决定吗?

As the video demonstrates, Jeff's chance of winning if he switches is $\begin{aligned} \frac{2}{3} \end{aligned}$ . His chance of winning if he doesn't switch is $\begin{aligned} \frac{1}{3} \end{aligned}$ . This can be extremely counter-intuitive . It helps to remember that at the beginning, Jeff has a $\begin{aligned} \frac{1}{3} \end{aligned}$ chance of choosing the correct door. Since the host knows where the car is and will always open a door with a goat behind it, the other door (the unopened door that Jeff did not originally choose) will have the car behind it $\begin{aligned} \frac{2}{3} \end{aligned}$ of the time. A good strategy for winning the car in this game is to always switch doors when given the choice.
::视频显示,如果Jeff开关是23岁,Jeff赢的机会就是23岁,如果他不开关,他赢的机会就是13岁,这可能是极反直觉的。这可以帮助人们记住,在一开始,Jeff有13个选择正确门的机会。因为东道主知道汽车在哪里,而且总是会用山羊打开门,另一扇门(Jeff最初没有选择的未开的门)就会有23岁。在这场比赛中赢得汽车的好策略是,在做出选择时总是开门。

Example 2
::例2

Use the random number generator to randomly choose 10 numbers between 1 and 100.
::使用随机数字生成器随机选择 1 到 100 之间的 10 个数字。

On your calculator, use the randInt( function. Enter randInt(1, 100, 10) and record the 10 numbers that come up. Numbers will vary depending on where you set your calculator to start.
::在计算器上,使用 randInt (函数) 。输入 randInt (1, 100, 10) 并记录生成的 10 个数字。数字将随您设置计算器的起始位置而变化。

Example 3
::例3

Ben makes a game involving tossing three coins and noting the sequence of heads and tails. If more heads than tails appear then he wins. If more tails than heads appear then you win. Is this game fair?
::Ben做了一个游戏,涉及抛掷三枚硬币,并记下头和尾的顺序。如果出现比尾巴更多的头,他就会赢。如果出现比尾巴多的尾巴,你就会赢。这个游戏公平吗?

To see if the game is fair, calculate Ben's chance of winning and your chance of winning. For the experiment of tossing three coins, the sample space is $\begin{aligned} {H H H, H H T, H T H, T H H, H T T, T H T, T T H, T T T} . \end{aligned}$ The probability of Ben winning is $\begin{aligned} \frac{4}{8} = \frac{1}{2} \end{aligned}$ because 4 of the 8 outcomes involve more heads than tails . The probability of you winning is $\begin{aligned} \frac{4}{8} = \frac{1}{2} \end{aligned}$ because 4 of the 8 outcomes involve more tails than heads . This is a fair game .
::看看游戏是否公平, 请计算本的胜算机会和你的胜算机会。在试打三个硬币时, 样本空间是 {HH, HHT, HTH, HH, HTT, TTH, TTT} 。 Ben获胜的概率是 48=12, 因为8个结果中, 4个涉及头部而不是尾部。你胜的概率是 48=12, 因为8个结果中, 4个涉及的尾部多于头部。这是一个公平的游戏。

Example 4
::例4

What if Ben instead tosses four coins? If no heads or an even number of heads appears then he wins. If an odd number of heads appears then you win. Is this game fair?
::如果Ben扔了四个硬币呢?如果没有头或甚至几个头,他就会赢。如果出现一个奇数的头,你就会赢。这公平吗?

The sample space has 16 outcomes: $\begin{aligned} {H H H H, H H H T, H H T H, H T H H, T H H H, H H T T, H T H T, T H H T, \end{aligned}$ $\begin{aligned} T H T H, T T H H, H T T H, T T T H, T T H T, T H T T, H T T T, T T T T} . \end{aligned}$ The probability of Ben winning is $\begin{aligned} \frac{8}{16} = \frac{1}{2} \end{aligned}$ because 8 of the 16 outcomes involve no heads or an even number of heads . The probability of you winning is $\begin{aligned} \frac{8}{16} = \frac{1}{2} \end{aligned}$ because 8 of the 16 outcomes involve odd number of heads . This is a fair game .
::样本空间有 16 个结果 : {HHH, HHHT, HHT, HHTH, HHTH, HHTHH, HHTT, HHTHT, HHTH, HTH, TTH, TTTH, TTTH, TTHT, HTTTT, HTTT, HTTTT, TTTT} 。 Ben 获胜的概率是 816=12, 因为16 个结果中有8个没有头或偶数的头。您获胜的概率是 816=12, 因为 16 个结果中有8个涉及奇数的头。这是一个公平的游戏。

CK-12 PLIX Interactive
::CK-12 PLIX 交互式互动

Remember this!

A fair event in probability means that each outcome has an equal chance of happening.
::公平的概率事件意味着每个结果都有同样的机会发生。

Finding probabilities can help to analyze the fairness of games and make fair choices.
::寻找概率可以帮助分析游戏的公平性和做出公平的选择。

Review
::审查审查审查审查

1. What makes a game fair?
::1. 是什么使游戏公平?

2. Why do people sometimes "flip a coin" to make a decision?
::2. 人们为什么有时 " 翻硬币 " 来做决定?

3. You have one prize and 30 people that could win the prize. Describe at least 2 ways to fairly choose who gets the prize.
::3. 你有一个奖项,30人可以赢得奖项,说明至少两个公平选择谁获得奖项的途径。

4. You and your three siblings are trying to decide who gets the last cookie. Describe a way you could use coins to help make a fair decision about who gets the cookie.
::4. 你和你的三个兄弟姐妹正在试图决定谁得到最后的饼干,描述你可以用硬币来帮助作出关于谁得到饼干的公平决定的方法。

5. Use a random number generator to pick 8 numbers at random between 1 and 300.
::5. 使用随机数字生成器在1至300之间随机抽取8个数字。

6. Describe a situation where you might want to use a random number generator.
::6. 描述您可能想要使用随机数字生成器的情形。

7. Your friend invents a dice game. You roll two dice. If the sum of the numbers that show up is even, you win. If the sum of the numbers that show up is odd, he wins. Is this game fair? Explain.
::7. 你的朋友发明了骰子游戏,你掷了两骰子。如果数字的总和是偶数,你赢了。如果数字的总和是奇数,他赢了。这个游戏公平吗?解释。

8. Your friend changes the dice game. Now, if the sum of the numbers is 6 or less, you win. If the sum of the numbers is 7 or more, he wins. Is this game fair? Explain.
::8. 你的朋友改变了骰子游戏,现在,如果数字的总和是6或6以下,你赢了,如果数字的总和是7或7以上,他赢了,这个游戏公平吗?解释。

9. You and your two friends Shelly and Lisa each have a spinner like the one below. You make up a game where everyone spins their spinner. You win if everyone gets a different color. Shelly wins if everyone gets the same color. Lisa wins if exactly two people get the same color. Analyze this game. Is it fair? If not, who has the advantage?
::9. 你和你的两个朋友Shelly和Lisa都有一个像下面那样的旋转器。你设计了一个每个人都可以旋转的游戏。如果每个人都有不同的颜色,你就赢了。如果每个人都有相同的颜色,Shelly赢了。如果两个人有相同的颜色,Lisa赢了。分析一下这个游戏。公平吗?如果不是,谁有优势?

10. Paul's game is you toss three coins and win if you get exactly two heads. Steve's game is you toss four coins and win if you get exactly two heads. Whose game should you play to have a better chance of winning? Explain.
::10. 保罗的游戏是,你扔三枚硬币,如果你有两把脑袋,就赢。史蒂夫的游戏是,你扔四枚硬币,如果你有两把脑袋,就赢。你应该玩谁的游戏,才能有更好的胜算机会?解释。

11. You spin the spinner below three times. You win the game if you get purple at least once. Should you play the game? Explain.
::11. 将旋转器旋转到3次以下;至少得到一次紫色,即可赢得比赛;应否玩?解释。

12. Deb makes up a card game. You draw two cards from a deck. If the cards are the same color (both red or both black) then she wins. If the cards are different colors (one is red and one is black) then you win. Is this game fair? Explain.
::12. Deb 编造一个牌游戏。你从牌牌上画两张牌。如果牌的颜色相同( 红或黑), 她就会赢。如果牌的颜色不同( 一个是红色, 一个是黑色), 你就会赢。这个游戏公平吗? 请解释。

13. Rachel makes up another card game. You draw one card from a deck. If the card is a spade, a diamond, or a jack, then Rachel wins. Otherwise, you win. Is this game fair? Explain.
::13. Rachel 编造了另一场牌游戏,从甲板上抽一张牌,如果牌是纸笔、钻石或杰克,Rachel就会赢,否则你赢,这公平吗?解释。

14. You toss two coins and roll a die. If the coins match and the die is an even number, you win. If the coins don't match and the die is greater than two, your friend wins. If anything else happens, nobody wins. Is this game fair? Explain.
::14. 你抛出两枚硬币,然后滚死。如果硬币匹配,而死亡是偶数,你就会赢。如果硬币不匹配,死亡则超过两枚,你朋友会赢。如果发生其他事情,没有人赢。这个游戏公平吗?请解释。

15. Gerry goes on a game show similar to the one Jeff was one (from the Concept problem). This time, there are 4 doors and a car is behind one of them. Gerry will have to pick a door. Then, the host will open two of the other doors to reveal goats and ask Gerry if he wants to switch his choice. He says his strategy will be to switch when he plays the game. Is this a good strategy? Explain.
::15. 格里参加一个类似Jeff(概念问题)的游戏节目,这次有四扇门,一辆汽车在其中一辆后面。Gerry必须开一扇门。然后,东道主将打开另外两扇门,以露出山羊,并询问Gerry是否愿意改变选择。他说,他的策略是当他玩游戏时转换。这是一个很好的策略吗?请解释。

16. A bag contains 5 yellow marbles, 6 blue marbles, and 4 white marbles. What is the probability that a marble chosen at random will not be white? If you do not choose white, you win. Is the game fair? Explain.
::16. 袋内装有5个黄色大理石、6个蓝色大理石和4个白色大理石,随机选择的大理石不会是白色的可能性有多大?如果不选择白色大理石,你就会赢。游戏公平吗?解释一下。

17. Ervin was given a choice for his test. There were 6 blue balls in a bag as well as 4 red balls. If he chose a blue ball, he would have to solve a trigonometry problem. If he chose a red ball, he would choose a geometry proof to do. What is the probability that Ervin will choose a blue ball? Is the game fair? Explain.
::17. Ervin被给予考试选择权:一个袋子里有6个蓝球,4个红色球,如果他选择一个蓝色球,他必须解决三角测量问题;如果他选择一个红色球,他将选择一个几何学证明,Ervin选择一个蓝色球的可能性有多大?这个游戏公平吗?解释一下。

Review (Answers)
::审查(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

Fairness ::公平公正

Probability ::概率

Examples ::实例实例实例实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

CK-12 PLIX Interactive ::CK-12 PLIX 交互式互动

Review ::审查审查审查审查

Review (Answers) ::审查(答复)