Course: 11.7 理论概率 | The Way To Learn

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理论概率
Jessie has a spinner that is divided into four colors: blue, red, yellow and green. She knows that the spinner can be used to calculate theoretical probability , but she can’t remember what theoretical probability is. She is wondering how it is different from experimental probability. Can you help her by giving her an answer?
::Jessie有一个可分为四种颜色的旋转器:蓝色、红色、黄色和绿色。她知道旋转器可以用来计算理论概率,但她不记得理论概率是什么。她想知道这与实验概率有什么不同。你能给她一个答案来帮助她吗?

In this concept, you will learn how to define and calculate theoretical probability.
::在此概念中,您将学会如何定义和计算理论概率。

Theoretical Probability
::理论概率

Probability is defined as a mathematical way of calculating how likely an event is to occur. The probability of an event occurring is defined as the ratio of favorable outcomes to the number of possible equally likely total outcomes in a given situation.
::概率的定义是计算事件发生可能性的一种数学方法,事件发生概率的定义是,在特定情况下,有利结果与可能同样可能的总结果数目之比。

In ratio form, the probability of an event is:
::在比率形式上,事件发生的概率是:

$\begin{aligned} P (event) = # of favorable outcomes : total # of outcomes \end{aligned}$
$\begin{align*}P(\text{event})= \# \ \text{of favorable outcomes}: \text{total}\ \# \ \text{of outcomes}\end{align*}$
::P(活动)+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++#####+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++###########++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Theoretical probability is probability that is based on an ideal situation.
::理论概率是建立在理想状态基础上的概率。

For instance, since a flipped coin has two sides and each side is equally likely to land up, the theoretical probability of landing heads (or tails) is exactly 1 out of 2. Whether or not the coin actually lands on heads (or tails) 1 out of every 2 flips in the real world does not affect theoretical probability. The theoretical probability of an event remains the same no matter how events turn out in the real world.
::例如,由于翻硬币有两面,而每面都同样有可能着陆,因此,着陆头(或尾巴)的理论概率完全为2分之一。无论硬币是否实际落在现实世界中每2个翻转中1个头(或尾巴)上,都不会影响理论概率。无论现实世界中发生什么事件,事件理论概率都保持不变。

Let’s look at an example.
::让我们举个例子。

Find the probability of tossing a number cube and getting a 4.
::找到抛掷一个数字立方体和获得一个四的概率。

First, find the total number of outcomes
::首先,找到成果的总数

Outcomes: 1, 2, 3, 4, 5, and 6
::结果:1、2、3、4、5和6

Total outcomes: 6
::成果共计:6

Second, find the number of favorable outcomes.
::第二,找到有利结果的数量。

Favorable outcomes:
::有利结果:

$\begin{align*}\text{Getting a } 4=1 \text{ favorable outcome}\end{align*}$
::获得4=1的有利结果

Then, find the ratio of favorable outcomes to total outcomes.
::然后,找到有利结果与总结果的比例。

$\begin{aligned} \begin{array}{rcl} P (event) & = & # of favorable outcomes : total # of outcomes \\ P (4) & = & 1 : 6 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} P(\text{event}) &=& \# \ \text{of favorable outcomes}: \text{total}\ \# \ \text{of outcomes} \\ P(4) &=& 1:6 \end{array}\end{align*}$
::P(活动)_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

A prediction is a reasonable guess about what will happen in the future. Good predictions should be based on facts and probability.
::预测是对未来会发生什么的合理猜测。良好的预测应该基于事实和概率。

Predictions based on theoretical probability. These are the most reliable types of predictions, based on physical relationships that are easy to see and measure and that do not change over time. They include such things as:
::基于理论概率的预测。这些预测是最可靠的预测类型,基于易于看到和测量且不会随时间变化的物理关系。这些预测类型包括:

coin flips
::硬硬币翻翻

spinners
::圆柱形

number cubes
::数字立方体数

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Jessie and the spinner.
::早些时候,你得到一个问题关于杰西和旋翼手。

The spinner can be used to calculate theoretical probability when it is considered without an experiment . Looking at the spinner itself, you can calculate the chances of spinning one or more colors. You haven’t done an experiment with the spinner yet, you are simply calculating the probability by looking at the spinner.
::旋转器可以用来计算没有实验就考虑的理论概率。查看旋转器本身, 您可以计算旋转一种或多种颜色的概率。您还没有对旋转器做实验, 您只是通过查看旋转器来计算概率。

The spinner has four colors so the total number of outcomes is 4. If she wanted to know the theoretical probability of spinning and getting on blue it would be 1:4 since there is only one section that is blue.
::旋转器有四种颜色,所以结果总数是4。如果她想知道旋转和升入蓝色的理论概率, 那就是1:4, 因为只有一个部分是蓝色的。

Experimental probability can be calculated once the spinner has been spun once or a number of times.
::一旦旋转器被转动一次或多次,即可计算出实验概率。

Example 2
::例2

Two number cubes are thrown. What is the theoretical probability of rolling a number greater than 8?
::抛出两个数字立方体。滚动数大于8的理论概率是多少?

First, let’s think about the different combinations that you can possibly roll on the number cubes. If you roll a 1 and a 2 for instance, you get a sum of 3. Look at the table below to show all of the combinations.
::首先,让我们想想您可以在数字立方体上滚动的不同组合。比如,如果你滚动一和二,你就会得到三下的总和。看看下面的表格,显示所有的组合。

The total number of outcomes is 36.
::结果总数为36项。

Next, the favorable outcomes are shaded in yellow in the table above.
::其次,有利结果在上表以黄色阴影显示。

$\begin{align*}\# \ \text{favorable outcomes }= 10\end{align*}$
::# 有利的结果=10

Then, find the ratio of favorable outcomes to total outcomes.
::然后,找到有利结果与总结果的比例。

$\begin{aligned} \begin{array}{rcl} P (event) & = & # of favorable outcomes : total # of outcomes \\ P (> 8) & = & 10 : 36 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} P(\text{event}) &=& \# \ \text{of favorable outcomes}: \text{total}\ \# \ \text{of outcomes} \\ P(>8) &=& 10:36 \end{array}\end{align*}$
::P(活动)++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Then, simplify the ratio.
::那么,简化比例。

$\begin{aligned} \begin{array}{rcl} P (> 8) & = & 10 : 36 \\ P (> 8) & = & 5 : 18 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} P(>8) &=& 10:36 \\ P(>8) &=& 5:18 \end{array}\end{align*}$
::P( > 8) = 10: 36P( > 8) = 5: 18

The answer is 5:18.
::答案是5点18分

The theoretical probability of rolling a sum greater than 8 when rolling two number cubes is 5:18.
::当滚动两个数字立方体时滚动大于8的数值的理论概率为5:18。

Calculate each example of theoretical probability.
::计算每个理论概率的例子。

Example 3
::例3

What is the probability of tossing a number cube and having it come up a two or a three?
::抛掷一个数字立方体,然后产生一个2或3的概率是多少?

First, find the total number of outcomes
::首先,找到成果的总数

Outcomes: 1, 2, 3, 4, 5, and 6
::结果:1、2、3、4、5和6

Total outcomes: 6
::成果共计:6

Next, find the number of favorable outcomes.
::接下来,找出有利结果的数目。

Favorable outcomes:
::有利结果:

$\begin{aligned} Getting a 2 or a 3 = 2 favorable outcomes \end{aligned}$
$\begin{align*}\text{Getting a }2 \text{ or a }3 = 2 \text{ favorable outcomes}\end{align*}$
::获得2或3=2的有利结果

Then, find the ratio of favorable outcomes to total outcomes.
::然后,找到有利结果与总结果的比例。

$\begin{aligned} \begin{array}{rcl} P (event) & = & # of favorable outcomes : total # of outcomes \\ P (2 or 3) & = & 2 : 6 \\ P (2 or 3) & = & 1 : 3 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} P(\text{event}) &=& \# \ \text{of favorable outcomes}: \text{total}\ \# \ \text{of outcomes} \\ P(2 \ \text{or} \ 3) &=& 2:6 \\ P(2 \ \text{or} \ 3) &=& 1:3 \end{array}\end{align*}$
::P(活动)++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

The answer is 1:3.
::答案是1:3。

The theoretical probability of rolling a 2 or a 3 on a number cube is 1:3.
::在数字立方体上滚动2或3的理论概率为1:3。

Example 4
::例4

What is the probability of tossing a number cube and having it come up even?
::抛掷一个数字立方体并使其达到平衡的概率是多少?

First, find the total number of outcomes
::首先,找到成果的总数

Outcomes: 1, 2, 3, 4, 5, and 6
::结果:1、2、3、4、5和6

Total outcomes: 6
::成果共计:6

Next, find the number of favorable outcomes.
::接下来,找出有利结果的数目。

Favorable outcomes:
::有利结果:

$\begin{aligned} \begin{array}{rcl} Getting an even number & = & 2, 4, 6 \\ = & 3 favorable outcomes \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \text{Getting an even number} &=& 2, 4, 6\\ &=& 3 \text{ favorable outcomes} \end{array}\end{align*}$
::获得偶数=2,4,6=3的有利结果

Then, find the ratio of favorable outcomes to total outcomes.
::然后,找到有利结果与总结果的比例。

$\begin{aligned} \begin{array}{rcl} P (event) & = & # of favorable outcomes : total # of outcomes \\ P (event #) & = & 3 : 6 \\ P (event #) & = & 1 : 2 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} P(\text{event}) &=& \# \ \text{of favorable outcomes}: \text{total}\ \# \ \text{of outcomes} \\ P(\text{event}\ \#) &=& 3:6 \\ P(\text{event}\ \#) &=& 1:2 \end{array}\end{align*}$
::P(活动)++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

The answer is 1:2.
::答案是1:2。

The theoretical probability of rolling an even number on a number cube is 1:2.
::在数字立方体上滚动偶数的理论概率为1:2。

Example 5
::例5

What is the probability of tossing a number cube and having it come up less than 6?
::抛掷一个数字立方体并让它上升小于6的概率是多少?

First, find the total number of outcomes
::首先,找到成果的总数

Outcomes: 1, 2, 3, 4, 5, and 6
::结果:1、2、3、4、5和6

Total outcomes: 6
::成果共计:6

Second, find the number of favorable outcomes.
::第二,找到有利结果的数量。

Favorable outcomes:
::有利结果:

$\begin{aligned} \begin{array}{rcl} A number less than 6 & = & 1, 2, 3, 4, 5 \\ = & 5 favorable outcomes \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \text{A number less than }6 &=& 1, 2, 3, 4, 5\\ &=& 5 \text{ favorable outcomes} \end{array}\end{align*}$
::数量小于6=1,2,3,4,5=5的有利结果

Then, find the ratio of favorable outcomes to total outcomes.
::然后,找到有利结果与总结果的比例。

$\begin{aligned} \begin{array}{rcl} P (event) & = & # of favorable outcomes : total # of outcomes \\ P (# < 6) & = & 5 : 6 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} P(\text{event}) &=& \# \ \text{of favorable outcomes}: \text{total}\ \# \ \text{of outcomes} \\ P(\# < 6) &=& 5:6 \end{array}\end{align*}$
::P( 活动) ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

The answer is 5:6.
::答案是5点6分

The theoretical probability of rolling a number less than 6 on a number cube is 5:6.
::在数字立方体上滚动数字小于6的理论概率为5:6。

Review
::回顾

Solve each problem.
::解决每一个问题。

A spinner has five sections: purple, yellow, green, blue and red.
::脊柱有五个部分:紫色、黄色、绿色、蓝色和红色。

1. Find the probability for the arrow landing on blue on the spinner:
::1. 发现箭头在旋转器上坠落于蓝色的概率:

List each favorable outcome.
::列出每个有利结果。

Count the number of favorable outcomes.
::计算有利结果的数目。

Write the total number of outcomes.
::记录结果的总数。

Write the probability.
::写下概率。

2. Find the probability for the arrow landing on red or green on the spinner:
::2. 发现箭头在旋转器上坠落红或绿的概率:

List each favorable outcome.
::列出每个有利结果。

Count the number of favorable outcomes.
::计算有利结果的数目。

Write the total number of outcomes.
::记录结果的总数。

Write the probability.
::写下概率。

3. Find the probability for the arrow NOT landing on yellow on the spinner:
::3. 发现箭头不降落在旋转器上黄色上的概率:

List each favorable outcome.
::列出每个有利结果。

Count the number of favorable outcomes.
::计算有利结果的数目。

Write the total number of outcomes.
::记录结果的总数。

Write the probability.
::写下概率。

4. Find the probability for rolling a 3 or 4 on the number cube:
::4. 在数字立方体上找到滚动3或4的概率:

List each favorable outcome.
::列出每个有利结果。

Count the number of favorable outcomes.
::计算有利结果的数目。

Write the total number of outcomes.
::记录结果的总数。

Write the probability.
::写下概率。

5. Find the probability for rolling greater than 2 on the number cube:
::5. 在数字立方体上找到滚动大于2的概率:

List each favorable outcome.
::列出每个有利结果。

Count the number of favorable outcomes.
::计算有利结果的数目。

Write the total number of outcomes.
::记录结果的总数。

Write the probability.
::写下概率。

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。

Section outline

General

理论概率

Theoretical Probability ::理论概率

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Calculate each example of theoretical probability. ::计算每个理论概率的例子。

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Theoretical Probability
::理论概率

Examples
::实例

Example 1
::例1

Example 2
::例2

Calculate each example of theoretical probability.
::计算每个理论概率的例子。

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)