计数事件

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Jessica is excited to attend her Halloween middle school dance.  There is no dress code, so her options for the outfit she will wear is wide open.  She has limited her choices to the following:
::Jessica很兴奋能参加万圣节中学舞蹈,

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Tops:  a lacy blouse or a graphic t-shirt
::顶部: 薄纱布或图形T恤衫

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Bottoms:  skinny jeans or an A-line skirt
::底底:紧身牛仔裤或A线裙子

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Shoes:  platform sandals or Vans tennis shoes
::鞋:平台凉鞋或凡斯网球鞋

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How many different outfits can Jessica create given these options?
::如果有了这些选择 杰西卡能创造多少不同的装备?

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In this concept, you will learn how to calculate the number of outcomes that exist when given different possibilities.
::在这个概念中,如果存在不同的可能性,你将学会如何计算结果的数量。

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Tree Diagrams
::树图图

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A good way to find the total number of outcomes for events is to make a tree diagram . A tree diagram is a branching diagram that shows all possible outcomes for an event . To make a tree diagram, split the different events into either-or choices. You can list the choices in any order.
::找到事件结果总数的一个好方法是绘制树图。树图是一个分支图,显示事件的所有可能结果。要绘制树图,可以将不同事件分成任意或任意选择。您可以按任何顺序列出选择。

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Let's look at an example.
::让我们举个例子。

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Nadia’s soccer team has 2 games to play this weekend. How many outcomes are there for Nadia’s team? Here is a tree diagram for game 1 and game 2.  Below that is a tree diagram illustrating the possible outcomes if Nadia were to play 3 games.
::Nadia的足球队本周末要玩两场比赛。 Nadia的球队有多少结果? 这是第1场和2场比赛的树图,下面是显示如果Nadia玩3场比赛可能的结果的树图。

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As you can see, there are four different outcomes for the two games:
::如你所见,两场比赛有四个不同的结果:

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win-win           win-lose              lose-win                 lose-lose
::双赢赢赢赢赢赢赢赢赢输赢赢输赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢赢

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If Nadia played 3 games, there would be 8 different outcomes:
::如果Nadia玩三场游戏,

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win-win-win                 win-lose-win                lose-win-win                   lose-lose-win
::双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢

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win-win-lose                win-lose-lose               lose-win-lose                  lose-lose-lose
::双赢 双赢 双赢 双赢 双赢 双赢 双输 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双赢 双输

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A tree diagram is a great way to visually see all of the options possible. It can also help you to organize the outcomes so that you don’t miss any possibilities.
::树图是观察所有可能的选项的好方法。 它也可以帮助您组织结果,以便您不会错失任何可能性。

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To remodel her kitchen, Gretchen has the following choices: Floor: tile or wood; Counter: Granite or formica; Sink: white, steel, stone. How many different choices can Gretchen make?
::为了改造厨房,格雷琴有以下选择:地板:瓷砖或木头;反面:格拉尼特或形体;辛克:白色、钢、石。格雷琴可以作出多少不同的选择?

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First, list the choices:
::首先,列出选择:

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Choice 1 Floor : tile, wood Choice 2 Counter : granite, formica Choice 3 Sink : white, steel, stone
::对应:花岗岩、形式选择3 辛克:白、钢、石

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Next, start a tree diagram by listing any of the options for Choice 1, branching off to options for Choice 2, and finally Choice 3. Make sure Choice 2 repeats for each branch of Choice 1, and that Choice 3 repeats for each branch of Choice 2.
::下一个,先列出选择1的任何选择,然后列出选择1的任何选择,然后细分为选择2的选择2,最后是选择3,确保选择2重复选择1的每个分支,选择3重复选择2的每个分支。

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Some spaces are left for you to fill in.
::有些空位留给您填充 。

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Then fill in the outcomes:
::然后填充结果:

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The answer is that there are 12 possible outcomes for the kitchen design.
::答案是厨房设计有12个可能的结果。

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While provide a visual layout of choice options and outcomes, they can take time to create.  Another quicker way to calculate the number of possible outcomes when provided different categories of choices is to multiply together the number of choice options for each category.
::在提供选择选项和结果的视觉布局的同时,这些选项和结果需要时间来创建。 计算提供不同选择类别时可能结果的数量的另一种快捷方式是将每一类别选择选项的数量相乘。

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For example, Kelly has chocolate or vanilla ice cream, three choices of toppings and four sauces.  How many possible outcomes does she have for creating her ice cream sundae?
::比如,凯莉有巧克力或香草冰淇淋,三个选择的托普和四个酱汁。她能创造多少冰淇淋圣代?

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$\begin{array}{r}2(icecream)x3(toppings)x4(sauces)=24possibleoutcomes\end{array}$
::2 (冰淇淋) x 3 3 (牙套) x 4 4 (酱) = 24 可能的结果

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Examples
::实例

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Example 1
::例1

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Earlier, you were given a problem about Jessica and her outfit for the middle school Halloween dance.
::早些时候,你得到一个问题 杰西卡 和她的服装 在中学万圣节舞蹈。

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She had limited her choices to a lacy blouse or a graphic t-shirt, skinny jeans or an A-line skirt, and platform sandals or Vans tennis shoes.  How many combinations can Jessica make:
::她只选择了拉西衬衫或图形T恤、紧身牛仔裤或A线裙子、平台凉鞋或Vans网球鞋。杰西卡可以做多少组合:

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First, list all of the choices and options for each:
::首先,列出每个选择和选项的所有选择和选项:

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Tops:  lacy blouse or graphic t-shirt
::顶部:薄纱布或图形T恤衫

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Bottoms:  skinny jeans or A-line skirt
::底底:紧身牛仔裤或A线裙子

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Shoes:  platform sandals or Vans tennis shoes
::鞋:平台凉鞋或凡斯网球鞋

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Next, list the number of options for each category:
::下一步,列出每一类别选项的数目:

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Tops:  2
::顶部: 2

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Bottoms:  2
::底底数: 2

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Shoes:  2
::鞋鞋:2

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Then, multiple these numbers together to get the number of possible outcomes:
::然后,将这些数字加在一起,以获得可能的结果数量:

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2 x 2 x 2 = 8
::2 x 2 x 2 = 8

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The answer is Jessica has 8 different outfit combinations from which to choose for the dance.
::答案是杰西卡有8种不同的组合 来选择跳舞

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Example 2
::例2

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A local pizza place offers a pizza with several options. On a basic pizza, you can choose one of each option. The options are a sourdough or whole wheat crust, mozzarella or feta cheese, and one of the following toppings: black olives, green olives, green pepper, garlic, pepperoni, ground beef, mushrooms, or onions, .
::本地的比萨饼店提供披萨,有几种选择。在基本比萨饼上,你可以选择每种选择。选择的办法是酸酸或整个小麦壳、马扎里拉(mozzarella)或乳酪(feta cheese),还有以下一种:黑橄榄、绿橄榄、绿胡椒、大蒜、辣椒、土牛肉、蘑菇或洋葱。

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Based on these possibilities, how many possible outcomes are there?
::基于这些可能性,有多少可能的结果?

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First, list all of the choices:
::首先,列出所有的选择:

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Crust: sourdough or whole wheat
::结壳:甜甜或全麦

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Cheese:  mozzarella or feta
::奶酪:马扎里拉乳酪或feta

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Toppings:  black olives, green olives, green pepper, garlic, pepperoni, ground beef, mushrooms, or onions
::托普:黑橄榄、绿橄榄、绿胡椒、大蒜、辣椒、土牛肉、蘑菇或洋葱

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Next, count the number of options for each category:
::下一步,计算每一类别选项的数目:

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Crust: 2
::结结壳: 2

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Cheese:  2
::茄子: 2

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Toppings:  8
::拖鞋:8

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Then, multiply these numbers together to get the number of possible outcomes:
::然后,将这些数字一起乘以获得可能的结果数量:

$\begin{array}{r}2\times 2\times 8=32\end{array}$

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The answer is there are  $\begin{array}{r}32\end{array}$  different pizza options.
::答案是,有32种不同的比萨饼选择。

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Example 3
::例3

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Jeff has five different pairs of socks and three pairs of shoes. How many possible combinations are there?
::杰夫有五双不同的袜子和三双鞋子

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First, list all of the choices:
::首先,列出所有的选择:

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Socks: 5 different pairs
::袜子:5对不同的袜子

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Shoes:  3 different pairs
::鞋子:3对不同的鞋子

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Next, list the number of options for each category:
::下一步,列出每一类别选项的数目:

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Socks: 5
::索克:5

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Shoes: 3
::鞋子:3

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Then, multiply these numbers together to get the number of possible outcomes:
::然后,将这些数字一起乘以获得可能的结果数量:

$\begin{array}{r}5\times 3=15\end{array}$

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The answer is there are 15 possible combinations of shoes and socks.
::答案是,有15种可能结合的鞋子和袜子。

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Example 4
::例4

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Jessie has three sweaters, two turtlenecks and three jackets. How many possible combinations are there?
::杰西有三件毛衣,两件紧身衣和三件夹克

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First, list all of the choices: 
::首先,列出所有的选择:

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3 sweaters
::3件毛衣

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2 turtlenecks
::2个海龟领

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3 jackets
::3件夹克

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Next, list the number of options for each category:
::下一步,列出每一类别选项的数目:

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Sweaters:  3
::喷雾者:3人

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Turtlenecks:  2
::海龟颈:2

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Jackets:  3
::夹克:3

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Then, multiply these numbers together to get the number of possible outcomes:
::然后,将这些数字一起乘以获得可能的结果数量:

$\begin{array}{r}3\times 2\times 3=18\end{array}$

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The answer is Jessie can make 18 different combinations.
::答案是杰西能做出18种不同的组合

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Review
::回顾

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Use tree diagrams to figure out the total number of possible outcomes for each situation.
::使用树形图来计算每一种情况可能产生的结果的总数。

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1. The Triplex Theater has 3 different movies tonight: Bucket of Fun, Bozo the Great, and Pickle Man. Each movie has an early and late show. How many different movie choices are there?
   ::《三重剧场》今晚有三部不同的电影:《嬉戏的巴克特》、《博佐大人物》和《小菜人》。每部电影都有早晚的节目。有多少不同的电影选择?
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2. Raccoon Stadium offers the following seating plans for football games:
   ::浣熊体育馆提供下列足球比赛座位计划:

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• lower deck, middle loge, or upper bleachers
  ::下甲板、中正对数或上层漂白板
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• center, side, end-zone
  ::中、 侧、 末区

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How many different kinds of seats can you buy?
::你能买到多少个不同的座位?

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3. Cell-Gel cell phone company offers the following choices:
   ::Cell-Gel手机公司提供以下选择:

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• Free internet plan or Pay internet plan
  ::免费互联网计划或付费互联网计划
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• 1200, 2000, or 3000 minutes
  ::1200,2000,或3000分钟
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• Premium or standard phone
  ::保险费或标准电话

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How many different kinds of plans can you get?
::你能得到多少种不同的计划?

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4. The e-Box laptop computer offers the following options.
   ::e-Box笔记本电脑提供以下选择。

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• Screen: small, medium, or large
  ::屏幕: 小型、中型或大型
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• Memory: standard 1 GB, extra 2 GB
  ::内存:标准1GB,额外2GB
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• Colors: pearl, blue, black
  ::颜色:珍珠、蓝、黑

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How many different combinations are there for options?
::选项有多少不同的组合?

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5. Jesse has four pairs of pants and three shirts. How many different outfits can he make?
   ::杰西有四条裤子和三件衬衫 他能做多少套不同的衣服?
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6. If Jesse adds three sweaters, how many different outfits can he make now?
   ::如果杰西加了三件毛衣 他现在能做多少件不同的衣服?
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7. Kyle has selected three different paint colors and two trim colors for his bedroom. How many different color combinations can he create?
   ::Kyle为卧室选择了三种不同的油漆颜色和两种修剪颜色。他能创造多少种不同的颜色组合?
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8. There are four teams playing in a tournament. Each team will play two sports. How many different combinations are there for teams and sports?
   ::比赛有四个队,每个队有两场比赛。球队和运动有几场不同的组合?
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9. Candice is selecting classes to take next year. The language classes available are: French, German and Spanish. The elective classes available are: Chorus, Gymnastics and Intro to Music. How many different options does Candice have if she needs to choose one language class and one elective class to take next year?
   ::Candice正在选择明年的班级。 可用的语言班是: 法语、 德语和西班牙语。 可用的选修班是: 合唱、 体操和音乐入门。 如果Candice需要选择一个语言班和明年选修一个选修班, 她有多少不同的选择?
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10. Candice's school has just added Sign Language to the available language classes. How many different options does Candice have now?
    ::Candice的学校刚刚在现有的语言班中增加了手语。 Candice现在有多少不同的选择?

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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 。

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Resources
::资源