Course: 11.12 承认独立和独立事件

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承认独立和独立事件
Do you know the difference between a dependent event and an independent event? If someone runs a 5 minute mile and someone else runs a 10 minute mile are these two events dependent on each other?
::如果有人跑5分钟跑,而有人跑10分钟跑,这两个事件是互相依存的吗?

In this concept, you will learn to recognize and dependent events.
::在此概念中,您将学会识别和依附事件。

Independent and Dependent Events
::独立和独立事件

When you work with probability , you can think about different events and how these events impact each other. Let’s look at an example.
::当您以概率工作时,您可以思考不同的事件以及这些事件相互影响的方式。让我们来举个例子。

Suppose there are two events:
::假设有两件事:

Event $\begin{align*}A\end{align*}$ : Toss 5 on the number cube
::事件 A: 在数字立方体上拖移 5

Event $\begin{align*}B\end{align*}$ : Spin blue on the spinner
::事件B:在旋盘上旋转蓝色

The probability of each of these events by itself is easy enough to compute. In general:
::每一事件本身的概率都很容易计算出来。

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

If this is the case, then you can write the following ratios for rolling a 5 and spinning blue.
::如果情况是这样,那么您可以写下滚动5和旋转蓝色的下列比率。

$\begin{aligned} \begin{array}{rcl} P (5) & = & \frac{# of favorable outcomes}{total # of outcomes} \\ P (5) & = & \frac{1}{6} \\ P (blue) & = & \frac{# of favorable outcomes}{total # of outcomes} \\ P (blue) & = & \frac{1}{4} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} P (5) &=& \frac{\# \ \text{of favorable outcomes}}{ \text{total }\# \ \text{of outcomes}} \\ P(5) &=& \frac{1}{6} \\ P \text{(blue)} &=& \frac{\# \ \text{of favorable outcomes}}{ \text{total }\# \ \text{of outcomes}} \\ P \text{(blue)} &=& \frac{1}{4} \end{array}\end{align*}$

These two events were performed with a spinner and a number cube. Does event $\begin{align*}A\end{align*}$ affect the probability of event $\begin{align*}B\end{align*}$ in any way? That is, does the number cube landing on 5 affect where the arrow lands in the spinner? If not, then the two events are said to be independent events.
::这两个事件都是用一个旋转器和一个数字立方体进行的。事件A是否以任何方式影响事件B的概率?也就是说,数字立方体降落在5上是否影响箭头降落在旋转器中的位置?如果没有,那么这两个事件据说是独立的事件。

If the outcome of one event has no effect on the outcome of a second event, then the two events are independent events.
::如果一个事件的结果对第二个事件的结果没有影响,那么这两个事件是独立的事件。

Events $\begin{align*}A\end{align*}$ and $\begin{align*}B\end{align*}$ above are independent events. No matter how the number cube turns up, its outcome does not affect the outcome of spinning the spinner.
::以上事件A和B是独立的事件。无论数字立方如何出现,其结果都不会影响旋转旋转器的结果。

Now let’s think about a different kind of example, one where the outcome of one event does impact the outcome of another event.
::现在让我们想一个不同的例子,一个事件的结果确实影响到另一个事件的结果的例子。

A bag has 3 red marbles, 4 blue marbles, and 3 green marbles. Irene pulls 1 green marble out of the bag. Does this change the probability that the next marble Irene pulls out of the bag will be green?
::包里有3个红色大理石、4个蓝色大理石和3个绿色大理石。艾琳从包里拿出了1个绿色大理石。这是否改变了艾琳从包里拿出下一个大理石的可能性?

First, find the number of favorable outcomes and the total number of outcomes for the first pick from the bag.
::首先,从包中找到首选的有利结果的数量和结果的总数。

$\begin{aligned} \begin{array}{rcl} # of favorable outcomes & = & # of green marbles \\ = & 3 \\ total # of outcomes & = & # of marbles \\ = & 3 red + 4 blue + 3 green \\ = & 10 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \# \ \text{of favorable outcomes} &=& \# \ \text{of green marbles} \\ &=& 3 \\ \text{total }\# \ \text{of outcomes} &=& \# \ \text{of marbles} \\ &=& 3 \ \text{red}+ 4 \ \text{blue} + 3 \ \text{green} \\ &=& 10 \end{array}\end{align*}$

Next, find the probability of choosing a green marble.
::接下来,找到选择绿色大理石的概率。

$\begin{aligned} \begin{array}{rcl} P (green) & = & \frac{# of favorable outcomes}{total # of outcomes} \\ P (green) & = & \frac{3}{10} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} P \text{(green)} &=& \frac{\# \ \text{of favorable outcomes}}{ \text{total }\# \ \text{of outcomes}} \\ P \text{(green)} &=& \frac{3}{10} \end{array} \end{align*}$

Then, for the second marble, there are now only 9 marbles left in the bag and only 2 of them are green. So the probability of pulling out a green marble for the second marble is now:
::第二大理石现在只剩下9大理石了,其中只有2大理石是绿色的。因此,第二大理石可能拔出绿色大理石的概率是:

$\begin{aligned} \begin{array}{rcl} P (green) & = & \frac{# of favorable outcomes}{total # of outcomes} \\ P (green) & = & \frac{2}{9} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} P \text{(green)} &=& \frac{\# \ \text{of favorable outcomes}}{ \text{total }\# \ \text{of outcomes}} \\ P \text{(green)} &=& \frac{2}{9} \end{array} \end{align*}$
::P(绿色)+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++29

Clearly, the first event affected the outcome of the second event in this situation. So the two events are NOT independent. In other words, they are dependent events.
::显然,第一个事件影响着第二个事件的结果。因此这两个事件并不独立。换句话说,它们是依附事件。

If the outcome of one event has an effect on the outcome of a second event, then the two events are dependent events.
::如果一个事件的结果对第二个事件的结果产生影响,那么这两个事件就属于附属事件。

Sometimes, you have and you have events that overlap and are not mutually exclusive . Two events $\begin{align*}A\end{align*}$ and $\begin{align*}B\end{align*}$ that cannot occur at the same time are mutually exclusive events . can occur at the same time.
::有时,你和你有重叠和不相互排斥的事件。两个不能同时发生的A和B事件是相互排斥的事件。可以同时发生。

Events $\begin{align*}R(\text{red})\end{align*}$ and event $\begin{align*}T(\text{top})\end{align*}$ are overlapping events or mutually exclusive events because both events share one outcome $\begin{align*}(\text{red} - \text{top})\end{align*}$ .
::R(red)事件和T(top)事件是重叠事件或相互排斥的事件,因为两个事件都有一个结果(red-top)。

To calculate the probability of overlapping events, list the sample space and find the favorable events.
::要计算重叠事件的概率, 请列出样本空间并找到有利的事件。

$\begin{aligned} r e d - t o p blue - top \\ red - bottom blue - bottom \end{aligned}$
$\begin{align*}& \mathbf{red - top} \qquad \ \text{blue}-\text{top}\\ & \text{red}-\text{bottom} \quad \text{blue}-\text{bottom}\end{align*}$
::红色蓝色-蓝色-顶层-底蓝色-底部

The probability of red-top is:
::红顶的概率是:

$\begin{aligned} \begin{array}{rcl} P (red - top) & = & \frac{# of favorable outcomes}{total # of outcomes} \\ P (red - top) & = & \frac{1}{4} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} P(\text{red}-\text{top}) &=& \frac{\# \ \text{of favorable outcomes}}{ \text{total }\# \ \text{of outcomes}} \\ P(\text{red}-\text{top}) &=& \frac{1}{4} \end{array} \end{align*}$
::P(red-top) +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++14++++++++++++++++++++++++++++++++++++++++++++++++++++++++++14+14+14++++14++++++++++++++14+++++++++++++14++++++++++++++++14++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the race and the two runners. Two runners are in a race. The first runner runs a 5 minute mile and the second runner runs a 10 minute mile.
::早些时候,有人给了你一个赛事和两名跑者的问题。有两名跑者正在比赛中。第一个跑者跑5分钟,第二个跑者跑10分钟。

The events are independent. The speed of one runner does not impact the speed of another runner.
::事件是独立的,一个跑者的速度不会影响另一个跑者的速度。

For a single toss of a number cube, what is the probability of event $\begin{align*}E(\text{even})\end{align*}$ and event $\begin{align*}S(4)\end{align*}$ both occurring?
::对于一个数字立方体的单次托盘,事件E(偶数)和事件S(4)发生的可能性是多少?

First, identify the overlapping outcomes of both events.
::首先,确定这两次活动的重叠结果。

$\begin{aligned} \begin{array}{rcl} E (even) & = & 2, 4, 6 \\ S (4) & = & 4 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} E(\text{even}) &=& 2, 4, 6 \\ S (4) &=& 4 \end{array}\end{align*}$
::E(偶数)=2,4,6S(4)=4

Next, find the total number of outcomes.
::接下来,找到结果的总数。

$\begin{aligned} \begin{array}{rcl} Total outcomes & = & 1, 2, 3, 4, 5, 6 \\ = & 6 total outcomes \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \text{Total outcomes} &=& 1, 2, 3, 4, 5, 6 \\ &=& 6 \ \text{total outcomes} \end{array} \end{align*}$
::结果共计=1,2,3,4,5,6=6

Then, find the probability of the overlapping events.
::然后,找到重叠事件的概率。

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

Identify whether the following events are independent or dependent.
::确定以下事件是独立事件还是依赖事件。

Example 2
::例2

Rolling a 1 and then rolling a 6 on a number cube.
::滚1 然后滚6 在数字立方体上滚6

The answer is these events are independent events because the rolling of a one does not affect the second roll and getting a six.
::答案是这些事件是独立事件,因为一个事件的滚动不影响第二卷和获得六份。

Example 3
::例3

A bag has a red and two blue marbles.
::包里有红色和两颗蓝色弹珠

First, one blue marble is drawn and then replaced.
::首先,绘制一个蓝色大理石,然后更换。

Then, a red one is drawn.
::然后,一个红色的画。

Are these two events dependent or independent?
::这两种事件取决于还是独立?

The answer is these events are independent events because the blue marble was replaced; it does not affect the outcome of drawing of red.
::答案是这些事件是独立的事件,因为蓝色大理石被替换;它不影响绘制红色的结果。

Example 4
::例4

A box contains 5 white and 6 red marbles. What is the probability of successfully drawing, in order, a red marble and then a white marble?
::A 框包含 5 个白色和 6 个红色大理石。成功绘制红色大理石和白色大理石的概率是多少?

The answer is these events are dependent events because the red marble was not replaced; the number of total outcomes will decrease from 11 to 10 when choosing the white marble.
::答案是这些事件是附属事件,因为红色大理石没有被替换;在选择白色大理石时,总结果将从11个减少到10个。

Review
::回顾

Write whether events A and B are dependent or independent.
::写下事件A和B是否依附或独立。

1. A : Doug flips a coin. B : Marlene chooses a card out of a deck.
::1. A:道格翻硬币。B:Marlene从甲板上选择一张牌。

2. A : In a bag with 5 white marbles and 5 black marbles, Sanjay pulls out a white marble. B : Without returning the marble to the bag, Sanjay pulls out a second marble.
::2. A:在装有5个白色大理石和5个黑色大理石的袋子中,Sanjay拔出1个白色大理石。 B:Sanjay拔出第二块大理石,但不把大理石还给袋子。

3. A : Eddie chooses the color blue for his new bike. B : Eddie chooses lasagne from the dinner menu.
::3. A:埃迪为他的新自行车选择蓝色颜色。 B:埃迪从晚餐菜单中选择拉萨涅。

4. A : The probability that it will rain tomorrow. B : The probability that the Red Wings hockey team will win their game tomorrow.
::4. A:明天下雨的概率。B:红翼曲棍球队明天赢得比赛的概率。

5. A : The probability that it will rain tomorrow. B: The probability that the baseball team will have a rain delay.
::5. A:明天雨的概率。 B:棒球队的雨速延迟的概率。

6. A : From a deck of cards, the probability of one player drawing a heart from the deck. B : On the next player’s turn, the probability of drawing another heart.
::6. A:从牌牌甲板上看,一个玩家从牌甲板上画出一颗心的概率。 B:下一个玩家轮到另一个玩家时,再画出另一颗心的概率。

7. A : The probability of a spinner landing on blue 6 times in a row. B : The probability of the spinner landing on blue on the next spin.
::7. A: 旋转器在蓝色上连续着陆6次的概率。 B: 旋转器在蓝色上下旋转的概率。

8. A : The probability of flipping a coin and having it come up heads. B : The probability of flipping it again and having it come up heads.
::8. A: 抛硬币并让硬币抬头的概率。 B:再抛硬币并让硬币抬头的概率。

9. A : The probability that it will snow tomorrow. B : The probability of having a snow day from school.
::9. A:明天下雪的概率。 B:放学放雪的概率。

10. A : The probability that it will be 90 degrees. B : The probability of enjoying a hot day at the beach.
::10. A:温度为90度的概率。 B:在海滩享受炎热一天的概率。

11. A : The probability that it will rain tomorrow. B : The probability of getting an A on a math test.
::11. A:明天下雨的概率。 B:数学测试获得A的概率。

12. A: The probability that the Rockies will be in the playoffs. B : The probability that the Rockies will win the World Series.
::12. A: 落基山脉在季后赛的概率。 B: 落基山脉赢得世界系列赛的概率。

13. A : The probability that tomorrow will be sunny. B : The probability that tomorrow will be a full moon.
::13. A:明天是晴天的概率。 B:明天是满月的概率。

14. A : The probability that tomorrow will be sunny. B : The probability that tomorrow will be cloudy.
::14. A:明天是晴天的概率。 B:明天是阴云的概率。

15. A : The probability that it will be cold today. B : The probability that it will be a full moon tomorrow.
::15. A:今天寒冷的概率。 B:明天是满月的概率。

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

承认独立和独立事件

Independent and Dependent Events ::独立和独立事件

Examples ::实例

Example 1 ::例1

Identify whether the following events are independent or dependent. ::确定以下事件是独立事件还是依赖事件。

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Independent and Dependent Events
::独立和独立事件

Examples
::实例

Example 1
::例1

Identify whether the following events are independent or dependent.
::确定以下事件是独立事件还是依赖事件。

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)