Course: 12.9 可行性补充和倍增规则

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关于概率的适增和倍增规则
Abigail wants to pick out a bouquet of balloons for her best friend's birthday. At the store, there is a jar with equal amounts of each of the following colors of balloons: yellow, green, and blue. She randomly pulls and replaces balloons three times so that she can look at the colors. What is the probability that Abigail will randomly choose one of each color over the three pulls?
::Abigail想为她最好的朋友的生日挑一束气球。在商店里, 有一罐子的气球的颜色相等: 黄色、绿色和蓝色。她随机拉了三次气球并替换了三次气球, 这样她就能看一看颜色。 Abigail在三次气球中随机选择每种颜色的可能性有多大 ?

In this concept, you will learn how to calculate the probability of one outcome from an event that occurs more than once.
::在此概念中, 您将学习如何计算一个事件不止一次发生的结果的概率。

Additive and Multiplicative Rules for Probability
::关于概率的适增和倍增规则

Finding the probability of an event is a matter of finding the ratio of favorable outcomes to total outcomes . For example, the sample space for a single coin flip has two outcomes: heads and tails. So the probability of getting heads on any single coin flip is:
::查找事件概率是找到有利结果与总结果之比的问题。例如, 单个硬币翻转的样本空间有两个结果: 头和尾。因此, 在任何单个硬币翻转上获得头的概率是 :

$\begin{aligned} P (heads) = \frac{f a v o r a b l e o u t c o m e s}{t o t a l o u t c o m e s} = \frac{1}{2} \end{aligned}$

::P(头)=有利成果

You can see that the sample space is represented by a number in the total outcomes. For example, if you had a spinner with four colors, the colors by name would be the sample space and the number four would be the total possible outcomes.
::您可以看到样本空间在总结果中以数字表示。例如, 如果您有一个有四种颜色的旋转符, 则按名称排列的颜色将是样本空间, 而数字四将是可能的总结果。

What about if you flipped a coin more than one time?
::如果你翻硬币不止一次呢?

To find the probability of a single outcome for more than one coin flip, use a tree diagram to find all possible outcomes in the sample space.
::要找到一个以上的硬币翻转结果的概率, 请使用树图在样本空间中查找所有可能的结果。

Then count the number of favorable outcomes within that sample space to find the probability.
::然后计算该样本空间内的有利结果数量,以找到概率。

For example, to find the probability of tossing a single coin twice and getting heads both times, first make a tree diagram to find all possible outcomes.
::例如,要找到两次抛掷一枚硬币和两次获得头的概率,首先绘制树图,以找到所有可能的结果。

The diagram shows there are 8 total outcomes and they are paired with first toss option and second toss option.
::图表显示,总结果有8个,并配有第一至下选项和第二至下选项。

Next pick out the favorable outcome–in this case, the outcome “heads-heads” is shown in red. You could have selected any of the favorable outcomes for the probability to be accurate.
::下一个选择有利结果 — — 在本案中,结果“头头”用红色显示。你本可以选择任何有利结果才能准确。

Then, write the ratio of favorable outcomes to total outcomes in the sample space.
::然后,写下在样本空间中,有利结果与总结果的比例。

$\begin{aligned} P (heads-heads) = \frac{f a v o r a b l e o u t c o m e s}{t o t a l o u t c o m e s} = \frac{1}{4} \end{aligned}$

::P(头头-头)=有利成果

You can see that since 1 of 4 outcomes is a favorable outcome , the probability of the coin landing on heads 2 times in a row is $\begin{aligned} \frac{1}{4} \end{aligned}$ . The answer is $\begin{aligned} \frac{1}{4} \end{aligned}$ .
::你可以看到,由于4个结果中的1个结果是一个有利的结果,硬币连续两次落在头部的概率是14个,答案是14个。

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

What is the probability of flipping a coin two times and getting two matching results–that is, either two heads or two tails?
::翻翻硬币两次和取得两个匹配结果的概率是多少? 也就是说,两个头还是两个尾?

First, create a tree diagram to see all options:
::首先,创建树图以查看所有选项 :

Next, pick out the favorable outcomes on the same tree diagram. They are shown in red.
::接下来,选择同一棵树图上的有利结果。它们以红色显示。

You can see that 2 of 4 total outcomes match.
::你可以看到4个总结果中的2个匹配。

Then, use the formula for probability of an event to calculate the probability:
::然后,使用事件概率的公式来计算概率:

$\begin{aligned} P (2 heads or 2 tails) = \frac{f a v o r a b l e o u t c o m e s}{t o t a l o u t c o m e s} = \frac{2}{4} = \frac{1}{2} \end{aligned}$

::P(2个头或2个尾尾)=有利结果

The answer is the probability of flipping two heads or two tails is 1:2.
::答案是翻两个头或两个尾的概率是1:2。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Abigail and her balloon bouquet.
::早些时候,你得到一个问题阿比盖尔和她的气球花束。

She randomly pulls and replaces balloons three times from a jar that contains equal numbers of yellow, green, and blue. What is the probability of her pulling one of each color after her last pull.
::她随机地从一个含有相同数量黄、绿和蓝色的罐子中拉出并更换三次气球。在她最后一次拉出之后,她拉出每种颜色之一的概率是多少。

First, create a tree diagram of the possible outcomes:
::首先,绘制一个可能结果的树图:

1st Pull: yellow
::第一次拉拉:黄色

2nd Pull: yellow green blue
::第二拉号:黄绿色蓝色

3rd Pull: yellow green blue yellow green blue yellow green blue
::第三拉线: 黄色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色蓝色蓝色绿色蓝色蓝色绿色蓝色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色

1st Pull: green
::第一拉:绿色

2nd Pull: yellow green blue
::第二拉号:黄绿色蓝色

3rd Pull: yellow green blue yellow green blue yellow green blue
::第三拉线: 黄色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色蓝色蓝色绿色蓝色蓝色绿色蓝色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色

1st Pull: blue
::第一次拉拉:蓝色

2nd Pull: yellow green blue
::第二拉号:黄绿色蓝色

3rd Pull: yellow green blue yellow green blue yellow green blue
::第三拉线: 黄色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色蓝色蓝色绿色蓝色蓝色绿色蓝色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色绿色

Next, list all possible outcomes:
::下面列出所有可能的结果:

yellow-yellow-yellow, yellow-yellow-green, yellow-yellow-blue, yellow-green-yellow, yellow-green-green, yellow-green-blue, yellow-blue-yellow, yellow-blue-green, yellow-blue-blue
::黄色、黄黄、黄、黄、黄、黄、黄、黄、黄、黄、绿、黄、绿、黄、绿、黄、绿、黄、蓝、黄、蓝、黄、蓝色、黄、蓝色、蓝色、蓝色、蓝色、黄色、蓝色、蓝色、蓝色

green-yellow-yellow, green-yellow-green, green-yellow-blue, green-green-yellow, green-green-green, green-green-blue, green-blue-yellow, green-blue-green, green-blue-blue
::绿黄、绿黄、绿黄、绿黄、绿黄、绿绿、绿黄、绿绿绿、绿绿绿、绿绿、绿绿蓝、绿蓝色、绿蓝色、绿蓝色、绿蓝色、绿蓝色、绿蓝色、绿蓝色、绿蓝色、绿蓝色、绿蓝色

blue-yellow-yellow, blue-yellow-green, blue-yellow-blue, blue-green-yellow, blue-green-green, blue-green-blue, blue-blue-yellow, blue-blue-green, blue-blue-blue
::蓝-黄-黄-黄、蓝-黄-绿、蓝-黄-绿、蓝-黄-蓝、蓝-绿-绿、蓝-绿、蓝-绿-绿、蓝-绿-蓝、蓝-蓝-蓝、蓝-蓝-蓝-蓝-黄、蓝-蓝色-绿、蓝色-蓝色-蓝色-蓝色-蓝色-蓝色

Then, count the number of favorable outcomes (one of each color balloon) and total outcomes, and use the probability of favorable outcome formula to figure out the probability:
::然后,计算有利结果(每个彩色气球一个)和总结果的数量,并使用有利结果公式的概率来计算概率:

Favorable outcome = 6 times
::有利结果=6倍

Total outcomes = 27
::总结果=27

$\begin{aligned} P (one of each color) = \frac{f a v o r a b l e o u t c o m e s}{t o t a l o u t c o m e s} = \frac{6}{27} \end{aligned}$

::P(每种颜色一个) = 有利结果总结果= 627

The answer is the probability that Abigail will choose one of each color is $\begin{aligned} \frac{6}{27} \end{aligned}$ .
::答案是 Abigail 选择每种颜色之一的概率是 627 。

Example 2
::例2

What is the probability of a win-win-win?
::双赢的可能性有多大?

First, create a tree diagram of the possible outcomes:
::首先,绘制一个可能结果的树图:

Next, count the number of times the favorable outcome occurs:
::接下来,计算好结果发生的次数 :

Win-Win-Win occurs 1 time.
::双 win-win发生一次。

Then, using the formula for probability of favorable outcomes, calculate the probability:
::然后,使用有利结果概率的公式计算概率:

$\begin{aligned} P (win-win-win) = \frac{f a v o r a b l e o u t c o m e s}{t o t a l o u t c o m e s} = \frac{1}{8} \end{aligned}$

::P(双赢双赢)=有利成果

There are eight possible outcomes for the teams.
::这些小组可能取得8项成果。

There is one option for a win-win-win in all three games.
::这三场比赛都有一个双赢的选项。

The answer is the probability of a win-win-win outcome is $\begin{aligned} \frac{1}{8} \end{aligned}$ .
::答案是双赢结果的概率是18。

Example 3
::例3

Look at the tree diagram. What is the probability of the outcome being heads and then tails?
::看看树图,结果的几率是正面还是尾巴?

First, locate the outcomes that are Heads-Tails.
::首先,找到头帆的结果。

Next, count the number of times this outcome occurs:
::下一步, 计数此结果发生的次数 :

1 time.
::1次。

Then, use the probability of favorable outcomes formula to calculate the probability:
::然后,使用有利结果公式的概率来计算概率:

$\begin{aligned} P (heads-tails) = \frac{f a v o r a b l e o u t c o m e s}{t o t a l o u t c o m e s} = \frac{1}{4} \end{aligned}$

::P(头尾尾)=有利成果

The answer is the probability of the outcome being heads and then tails is $\begin{aligned} \frac{1}{4} \end{aligned}$ .
::答案是结果的概率是正面,然后尾巴是14。

Example 4
::例4

Look at the tree diagram. What is the probability of tails then heads?
::看看树图尾巴和头的概率是多少?

First, locate the outcomes that are Tails-Heads.
::首先,找到"尾头"的结果

Next, count the number of times this outcome occurs:
::下一步, 计数此结果发生的次数 :

1 time
::1 小时 1 次

Then, use the probability of favorable outcomes formula to calculate the probability:
::然后,使用有利结果公式的概率来计算概率:

$\begin{aligned} P (tails-heads) = \frac{f a v o r a b l e o u t c o m e s}{t o t a l o u t c o m e s} = \frac{1}{4} \end{aligned}$

:零售头)=有利成果

The answer is the probability of the outcome being tails and then heads is $\begin{aligned} \frac{1}{4} \end{aligned}$ .
::答案是结果有可能是尾巴,然后头是14。

Review
::回顾

Answer each question. Use tree diagrams when necessary.
::回答每个问题,必要时使用树图。

What is the probability that the arrow of the spinner will land on red on a single spin?
::旋转器箭头投在红色上的一个旋转上的可能性有多大?

If the spinner is spun two times in a row, what is the probability that the arrow will land on red both times?
::如果旋转器一行两次旋转,箭头两次落下红的概率是多少?

If the spinner is spun two times in a row, what is the probability that the spinner will land on the same color twice?
::如果旋转器一行旋转两次,旋转器在同一颜色降落两次的概率是多少?

If the spinner is spun two times in a row, what is the probability that the arrow will land on red at least one time?
::如果旋转器一行被旋转两次,箭头至少落下一次红色的概率是多少?

If the spinner is spun two times in a row, what is the probability that the spinner will land on a different color both times?
::如果旋转器一行旋转两次,旋转器在两种不同颜色降落的概率是多少?

If the spinner is spun two times in a row, what is the probability that the arrow will land on blue or green at least one time?
::如果旋转器一行旋转两次,箭头至少一次降落在蓝色或绿色上的可能性有多大?

Two cards, the Ace and King of hearts, are taken from a deck, shuffled, and placed face down. What is the probability that a single card chosen at random will be an Ace?
::两张牌,红桃王牌和红桃王牌,被从甲板上拿下来,打乱了,然后把脸放倒。随机选择的单张牌是A牌的可能性有多大?

If one card is chosen from the 2-card stack above, then returned to the stack and a second card is chosen, what is the probability that both cards will be Kings?
::如果从上面的2张牌堆中选择了一张牌,然后返回堆叠,然后选择第二张牌,那么两张牌都是国王的概率是多少?

If one card is chosen from the 2-card stack above, then returned to the stack and a second card is chosen, what is the probability that both cards will match?
::如果从上面的2张牌堆中选择一张牌,然后返回堆叠,然后选择第二张牌,那么两张牌的匹配概率是多少?

If one card is chosen from the 2-card stack above, then returned to the stack and a second card is chosen, what is the probability that both cards NOT match?
::如果从上面的2张卡堆中选择了一张卡,然后返回到堆叠中,然后选择第二张卡,那么两张卡的不匹配概率是多少?

Look at the tree diagram and figure out each probability as a ratio.
::看看树图找出每个概率作为比例

What is the probability of the option tile, steel and granite?
::选择瓷砖、钢和花岗岩的概率是多少?

What is the probability of either tile, steel, granite or tile, granite, white?
::瓷砖、钢铁、花岗岩或瓷砖、花岗岩、白色的概率是多少?

What is the probability of Formica being in the option?
::Formica选择的可能性有多大?

What is the probability of tile and Formica being in the option?
::瓷砖和Formica的概率有多大?

What is the probability of white and Formica being in the option?
::白人和Formica的概率有多大?

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。

Resources
::资源

Section outline

General

关于概率的适增和倍增规则

Additive and Multiplicative Rules for Probability ::关于概率的适增和倍增规则

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Additive and Multiplicative Rules for Probability
::关于概率的适增和倍增规则

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源