Section: 概率的定义 | 12.14概率的定义

Section outline

Mrs. Woodfaulk uses colored balls to demonstrate the concept of probability . She places 1 red ball, 1 blue, 3 yellow balls, and 4 green balls in a bag and closes the bag. She places three stacks of colored tokens on her desk. One stack is red. One is yellow. And one stack is green. Then, Mrs. Woodfaulk tells the class that she is going to pull one ball out the bag, and she wants each student to guess what color the ball will be by choosing the same color token. How can Rachel write a statement of probability for each color ball?
::Woodfaurk夫人用彩色球来证明概率的概念。她把1个红色球、1个蓝色球、3个黄色球和4个绿色球放在一个袋子里, 并关上袋子。她把三堆彩色标牌放在她桌上。一堆是红色的。一堆是黄色的。一堆是绿色的。然后Woodfaurk夫人告诉班级, 她要从袋子里拿出一个球, 她想让每个学生通过选择相同的颜色符号来猜这个球会是什么颜色。 Rachel怎么写每个彩色球的概率声明?

In this concept, you will learn the definition of probability and how to write statements of probability.
::在此概念中,您将学习概率的定义以及如何写出概率说明。

Writing Statements of Probability
::概率书面陈述书

Probability is the measure of the likeliness that an event will occur. Probability is used all the time in real life situations. If you watch the weather in the morning you may hear the meteorologist talk about a 20% chance or rain or snow. In this case a percentage gives us the probability that it would rain. While there is a 20% chance that it will rain, there is an 80% chance that it won’t rain. All in all, you are still talking about probability.
::概率是事件会发生的可能性的尺度。概率在现实生活中一直使用。如果你在早上看天气, 你可能会听到气象学家谈论20%的概率或雨或雪。在这种情况下, 一定百分比的概率给了我们下雨的概率。虽然有20%的降雨概率,但80%的概率不会下雨。总的来说, 你仍然在谈论概率。

A ratio is used to calculate probability. A ratio is a way of comparing two quantities. With probability, you can compare the number of favorable outcomes to the amount of possible outcomes.
::使用一个比率来计算概率。一个比率是一种比较两个数量的方法。如果有概率,可以比较有利结果的数量和可能的结果的数量。

Here is a ratio.
::这是比例。

$\begin{aligned} P = \frac{# of Favourable Outcomes}{# of Possible Outcomes} \end{aligned}$

Notice that the ratio is in fraction form. This is one way to figure out the probability of an event happening.
::注意该比例是分数形式。这是计算事件发生概率的一种方法。

Here is an example to illustrate the concept of probability. As you read this example, think about the number of possible outcomes first. That is your denominator. Then calculate the number of favorable outcomes, your numerator.
::这里有一个示例来说明概率的概念。当你阅读这个示例时, 请先考虑可能的结果的数量。这是您的分母。然后计算有利的结果的数量, 您的分子。

Mark is rolling a number cube that is numbered 1 – 6. What are the chances that Mark will roll a 2?
::马克正在滚动一个数字立方体,编号是1 - 6. Mark会滚动一个2的可能性有多大?

First, determine the number of possible outcomes. Since the number cube is numbered 1 – 6, there are only 6 possible outcomes. That is the denominator.
::首先, 确定可能的结果数量。由于数字立方体编号为 1 - 6, 只有 6 个结果。这就是分母。

$\begin{aligned} P = \frac{# number of favourable outcomes}{6} \end{aligned}$

Next, consider the number of favorable outcomes. Since 2 is the only favorable outcome , there is only 1 favorable outcome. That is the numerator.
::接下来,请考虑有利结果的数量。由于2是唯一有利结果,只有1个有利结果。这就是分子。

$\begin{aligned} P = \frac{1}{6} \end{aligned}$

This is the probability of Mark rolling a 2.
::这是马克滚动 2 的概率。

Now, let’s look at one that is a little more complicated.
::现在,让我们来看看一个更复杂的问题。

Jessie spins the same number cube. She wants to spin an odd number. What are the chances that she will spin an odd number?
::Jessie旋转了同一个数字立方体。她想旋转一个奇数。她转一个奇数的几率有多大?

First, the number of possible outcomes did not change. It is still a 6.
::首先,可能的结果数量没有改变,仍然是6个。

$\begin{aligned} P = \frac{# of Favourable outcomes}{6} \end{aligned}$

Next, determine the number of favorable outcomes. The questioned asked about spinning an odd number. Counting from 1 – 6, there are three odd numbers. Therefore, the number of favorable outcomes is 3.
::接下来,决定有利结果的数量。被询问者询问如何旋转一个奇数。从 1 - 6 算起,有三个奇数。因此, 有利结果的数量是 3 。

$\begin{aligned} P = \frac{3}{6} or \frac{1}{2} \end{aligned}$

Then, simplify the probability if necessary.
::然后,如果有必要,简化概率。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Rachel and Mrs. Woodfaulk’s probability demonstration.
::更早之前, 你得知Rachel和Woodfaulk夫人的概率演示有问题。

Mrs. Woodfaulk placed 1 red ball, 1 blue, 3 yellow balls, and 4 green balls in the bag, and then she asked the students to choose a token that is the color of the ball she will randomly pull from the bag. The students in the first three rows chose their tokens, and most of them chose green tokens. A few chose yellow tokens. And none of them chose the red token. Rachel was planning to choose a yellow token, but as she waited in line, she began to rethink her decision.
::Woodfaulk夫人在袋子里放了一个红色的球,一个蓝色的,3个黄色的球,还有4个绿色球,然后她要求学生们选择一个标志,即她随机从袋子里拉出的球的颜色。头3行的学生们选择了他们的标志,他们中的大多数选择了绿色的标志。几个选择了黄色的标志。没有一个选择了红色的标志。Rachel计划选择一个黄色的标志,但当她排队等待的时候,她开始重新考虑她的决定。

Should Rachel reconsider her decision to choose a yellow token? How can she write a probability statement that will help her choose the most likely colored ball?
::瑞秋应该重新考虑她选择黄色标志的决定吗?她怎么能写一个概率说明来帮助她选择最有可能的彩色球呢?

First, determine the number of possible outcomes. Since there are a total of 9 colored balls in the bag, there are 9 possible outcomes. That is the denominator.
::首先, 确定可能的结果数量。由于包中共有9个彩色球, 可能有9个结果。这就是分母。

$\begin{aligned} P = \frac{}{9} \end{aligned}$

Next, determine which color ball will have the highest number of favorable outcomes. There are 4 green balls in the bag, so the number of favorable outcomes for green is 4. That is the numerator.
::下一步, 确定哪个颜色球的有利结果最多。包里有 4 个绿色球, 所以绿色的有利结果是 4 个, 这就是分子。

$\begin{aligned} P = \frac{4}{9} \end{aligned}$

The probability of Mrs. Woodfaulk pulling a green ball is four out of nine. The probability of pulling a yellow ball is 3 out of nine. And the probability of pulling a red ball or a blue ball out of the bag is 1 out of 9.
::Woodfaulk夫人拉绿色球的概率是九分之四。拉黄球的概率是九分之三。拉红球或蓝球出包的概率是九分之一。拉黄球的概率是九分之三。拉红球或蓝球出包的概率是九分之一。

So, Rachel should choose a green token because the probability is more likely that Mrs. Woodfaulk will pull a green ball from the bag.
::所以,瑞秋应该选择一个绿色的标志因为很可能 Woodfaulk夫人会从袋子里拉一个绿色的球

Example 2
::例2

Find the probability.
::找到概率。

Jake put eight colored squares into a bag. There are two reds, four yellows, one green and one blue.
::Jake把八个彩色方块放进袋子里有两个红色四个黄色一个绿色一个蓝色

What is the probability that Jake will not pull out a yellow or a red square?
::Jake不会拔出黄色或红色方形的概率是多少?

First, determine the number of possible outcomes. Since there are 8 colored squares in the bag, there are 8 possible outcomes. That is the denominator.
::首先,确定可能的结果数量。因为包里有8个彩色方形, 可能有8个结果。这就是分母。

$\begin{aligned} P = \frac{}{8} \end{aligned}$

Next, determine the number of favorable outcomes or all of the possibilities that are not yellow or red. This means you count the green and the blue squares. There is one green and one blue square, so the number of favorable outcomes is 2. That is the numerator.
::下一步,确定有利结果的数量或所有非黄色或红色的可能性。这意味着您计算绿色和蓝色方块。有一个绿色和一个蓝色方块, 所以有利结果的数量是 2 。这就是数字。

$\begin{aligned} P = \frac{2}{8} \end{aligned}$

Then, simplify the fraction by dividing the numerator and denominator by their greatest common factor, 2.
::然后,通过将分子和分母除以其最大共同系数2来简化分数。

$\begin{aligned} \begin{array}{rcl} 2 \div 2 & = & 1 \\ 8 \div 2 & = & 4 \end{array} \end{aligned}$

This becomes
::变成

$\begin{aligned} P = \frac{1}{4} \end{aligned}$

This is the answer.
::这就是答案。

Write a ratio to show the probability for each question below regarding this scenario.
::写一个比率,以显示以下关于这一假设情景的每个问题的概率。

Jake put eight colored squares into a bag. There are two reds, four yellows, one green and one blue.
::Jake把八个彩色方块放进袋子里有两个红色四个黄色一个绿色一个蓝色

Example 3
::例3

What is the probability of Jake pulling out a red cube?
::杰克拔出红色立方体的概率是多少?

First, determine the number of possible outcomes. Since there are 8 colored squares in the bag, there are 8 possible outcomes. That is the denominator.
::首先,确定可能的结果数量。因为包里有8个彩色方形, 可能有8个结果。这就是分母。

$\begin{aligned} P = \frac{}{8} \end{aligned}$

Next, determine the number of favorable outcomes. This means count the red squares. There are 2, so the number of favorable outcomes is 2. That is the numerator.
::下一步,决定有利结果的数量。这意味着计数红色方形。有 2 个, 有利结果的数量是 2 个, 这就是数字。

$\begin{aligned} P = \frac{2}{8} \end{aligned}$

Then, simplify the fraction by dividing the numerator and denominator by their greatest common factor, 2.
::然后,通过将分子和分母除以其最大共同系数2来简化分数。

$\begin{aligned} \begin{array}{rcl} 2 \div 2 & = & 1 \\ 8 \div 2 & = & 4 \end{array} \end{aligned}$

This becomes
::变成

$\begin{aligned} P = \frac{1}{4} \end{aligned}$

This is the answer.
::这就是答案。

Example 4
::例4

What is the probability of Jake pulling out a yellow cube?
::Jake拔出黄色立方体的概率是多少?

First, determine the number of possible outcomes. Since there are 8 colored squares in the bag, there are 8 possible outcomes. That is the denominator.
::首先,确定可能的结果数量。因为包里有8个彩色方形, 可能有8个结果。这就是分母。

$\begin{aligned} P = \frac{}{8} \end{aligned}$

Next, determine the number of favorable outcomes or yellow squares. There are four yellow squares, so the number of favorable outcomes is 4. That is the numerator.
::下一步, 确定有利结果或黄色方块的数量。有四个黄色方块, 因此有利结果的数量是 4 。这就是分子。

$\begin{aligned} P = \frac{4}{8} \end{aligned}$

Then, simplify the fraction by dividing the numerator and denominator by their greatest common factor, 4.
::然后,通过将分子和分母除以其最大共同系数4来简化分数。

$\begin{aligned} \begin{array}{rcl} 4 \div 4 & = & 1 \\ 8 \div 2 & = & 4 \end{array} \end{aligned}$

This becomes
::变成

$\begin{aligned} P = \frac{1}{2} \end{aligned}$

This is the answer.
::这就是答案。

Example 5
::例5

What is the probability of Jake pulling out a yellow or blue cube?
::Jake拔出黄色或蓝色立方体的概率是多少?

First, determine the number of possible outcomes. Since there are 8 colored squares in the bag, there are 8 possible outcomes. That is the denominator.
::首先,确定可能的结果数量。因为包里有8个彩色方形, 可能有8个结果。这就是分母。

$\begin{aligned} P = \frac{}{8} \end{aligned}$

Next, determine the number of favorable outcomes or the number of yellow and blue cubes. There are 4 yellow squares and 1 blue square, so the number of favorable outcomes is 5. That is the numerator.
::下一步, 确定有利结果的数量或黄色和蓝色立方体的数量。有 4 个黄色方块和 1 个蓝色方块, 因此有利结果的数量是 5 个, 这就是分子。

$\begin{aligned} P = \frac{5}{8} \end{aligned}$

This is the answer and it cannot be simplified.

::这是答案,不能简化。

Review
::回顾

Without looking, you pull out one stone from the bag. A bag has the following 10 colored stones in it. There are 2 red ones, 2 blue ones, 3 green ones, 1 orange one, and 2 purple ones.
::袋子里有以下10块彩色的石头。有2块红色的,2块蓝色的,3块绿色的,1块橙色的,2块紫色的。

Write a fraction to show the following probabilities based on the scenario above. Remember to simplify your answers.
::写入一个分数以显示基于以上情景的以下概率。记住要简化您的答案。

One orange stone
::一个橙色石头

A red stone
::红石

A green stone
::绿宝石

A yellow stone
::黄宝石

A blue stone or an orange one
::蓝石或橙石

A red one or a blue one
::红色的或蓝色的

A green one or an orange one
::绿色的或橙色的

A blue one or a green one
::蓝色的或绿色的

A blue one or a purple one
::蓝色或紫色

A purple one or a red one
::紫色的还是红色的

Not purple
::不是紫色

Not red
::不是红色

Not orange or purple
::不是橙色或紫色

Not red or purple
::不是红色或紫色

Not orange
::不是橙色

Review (Answers)
::回顾(答复)

Section outline

Writing Statements of Probability ::概率书面陈述书

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)

Writing Statements of Probability
::概率书面陈述书

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)