课程： 14.4 基本概率 | The Way To Learn

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选择节基本概率

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基本概率
Introduction
::导言

Assume someone in your neighborhood wins a lottery game on average once every 100 days. What is the probability that someone will win the lottery in the next 100 days?
::假设你附近的人每100天平均赢一场彩票比赛。在未来100天中,有人会赢彩票的几率有多大?

Probability
::概率

Probability is the chance of an event occurring. Simple probability is defined as the number of successful outcomes divided by the total number of outcomes, assuming all outcomes are equally likely. The notation $\begin{aligned} P (E) \end{aligned}$ is read "the probability of event $\begin{aligned} E \end{aligned}$ ."
::概率是事件发生的概率。简单概率的定义是成功结果的数量除以结果的总数, 假设所有结果都具有同等可能性。 P( E) 符号为“ 事件发生的可能性 E ” 。

$\begin{aligned} P (E) = \frac{# successful outcomes}{# possible outcomes} \end{aligned}$

::P(E) 成功结果# 可能结果

Probabilities can be represented with fractions, decimals, or percents. Since the number of possible outcomes is in the denominator, the probability is always between 0 and 1. A probability of 0 means the event will definitely not happen, while a probability of 1 means the event will definitely happen.
::概率可以用分数、小数或百分比表示。由于分母中含有可能的结果, 概率总是在0到1之间。 0 的概率意味着事件绝对不会发生, 而 1 的概率意味着事件绝对会发生。

$\begin{aligned} 0 \leq P (E) \leq 1 \end{aligned}$

::0P(E)1

The following video introduces probability and determining the probability of basic events:
::以下视频介绍概率并确定基本事件的概率:

The probability of something not happening is called the complement , $\begin{aligned} P^{C} or P^{'} \end{aligned}$ , and is found by subtracting the probability from 1.
::未发生事件的概率称为补充、PC或P`,从1中减去概率即可发现。

$\begin{aligned} P (E^{C}) = 1 - P (E) \end{aligned}$

::P(EC)=1-P(E)

You will often be looking at probabilities of two or more independent experiments. Experiments are independent when the outcome of one experiment has no effect on the outcome of the other experiment: for instance, drawing a card from a deck and then replacing that card back in the deck prior to drawing a 2nd card. Since the initial card was replaced back in the deck, the 2nd draw is not affected by the 1st draw. Experiments are dependent when the outcome of one experiment has an effect on the outcome of the other experiment. Now, suppose after one card is drawn from the deck, it is not replaced. As a result, the deck will have one fewer card, the 1st card drawn, when the 2nd card is drawn, which will have an effect on the probability of this 2nd experiment.
::您通常会看到两个或更多个独立实验的概率。当一个实验的结果对另一个实验的结果没有影响时, 实验是独立的。例如, 从甲板上画一张牌, 然后在第二张卡之前在甲板上替换该卡。由于最初的卡被换回甲板, 第二张牌不会受到第一张图的影响。当一个实验的结果对另一个实验的结果产生影响时, 实验是独立的。现在, 假设从甲板上画一张牌之后, 它不会被替换。因此, 甲板将少一张卡片, 当第二张卡绘制出来时, 第一张卡片将会对第二次试验的概率产生影响。

If there are two independent experiments, one with outcome $\begin{aligned} A \end{aligned}$ and the other with outcome $\begin{aligned} B \end{aligned}$ , then the probability of $\begin{aligned} A \end{aligned}$ and $\begin{aligned} B \end{aligned}$ is
::如果有两个独立实验,一个是结果A试验,另一个是结果B试验,那么A和B的概率是

$\begin{aligned} P (A a n d B) = P (A) \cdot P (B) . \end{aligned}$

::P(A和B)=P(A)_P(B)_P(B)_P(A)_P(B)_P(A)_P(B)_P(B)_P(A)_P(A)_P(B)_P(B)_P(A)_P(A)_P(B)_P(B)_P(B)_B)。

The probability of $\begin{aligned} A \end{aligned}$ or $\begin{aligned} B \end{aligned}$ is
::A或B的概率为

$\begin{aligned} P (A o r B) = P (A) + P (B) - P (A a n d B) . \end{aligned}$

::P(A或B)=P(A)+P(B)-P(A和B)。

Play, Learn, and Explore with Probability:
::与概率一起玩、学习和探索:

Examples
::实例

Example 1
::例1

If you are dealt one card from a 52-card deck, what is the probability that you are dealt a heart? What is the probability that you are dealt a 3? What is the probability that you are dealt the 3 of hearts?
::如果你被从52张牌牌牌牌牌牌牌上打出一张牌,你被打心脏的几率是多少?你被打3的几率是多少?你被打红心3的几率是多少?

Solution:
::解决方案 :

There are 13 hearts in a deck of 52 cards. %3D%5Cfrac%7B13%7D%7B52%7D%3D%5Cfrac%7B1%7D%7B4%7D">
$\begin{aligned} P (h e a r t) = \frac{13}{52} = \frac{1}{4} \end{aligned}$

::52张牌牌牌牌牌牌牌上有13个红心。 P( 心脏)=1352=14

There are 4 threes in the deck of 52.
$\begin{aligned} P (t h r e e) = \frac{4}{52} = \frac{1}{13} \end{aligned}$

::52. P(3)=452=113的甲板上有4个3个。

There is only 1 three of hearts.
$\begin{aligned} P (t h r e e a n d h e a r t) = \frac{1}{52} \end{aligned}$

::红心只有13个。P(3和心脏)=152

Example 2
::例2

Dean and his friend Randy like to play a special poker game with their friends. Dean goes home a winner 60% of the time, and Randy goes home a winner 75% of the time.
::Dean和他的朋友Randy喜欢和他们的朋友玩特别的扑克牌游戏。Dean60%的时间是赢家回家,75%的时间是赢家回家。

1) What is the probability that they both win on the same night?
::1) 他们在同一晚获胜的概率有多大?

2) What is the probability that Randy wins and Dean loses?
::2)兰迪和迪安输赢的可能性有多大?

3) What is the probability that they both lose?
::3) 他们两人失去的可能性有多大?

Solutions:
::解决办法:

First represent the information with probability symbols.
::首先代表带有概率符号的信息。

Let $\begin{aligned} D \end{aligned}$ be the event that Dean wins. Let $\begin{aligned} R \end{aligned}$ be the event that Randy wins. The complement of each probability is when Dean or Randy loses instead.
::让D成为Dean获胜的事件。让R成为Randy获胜的事件。每种可能性的补充是当Dean或Randy输的时候。

$\begin{aligned} P (D) = 0.60, P (D^{C}) = 0.40 \end{aligned}$

::P(D)=0.60,P(DC)=0.40

$\begin{aligned} P (R) = 0.75, P (R^{C}) = 0.25 \end{aligned}$

::P(R)=0.75,P(RC)=0.25

1) $\begin{aligned} P (D a n d R) = P (D) \cdot P (R) = 0.60 \cdot 0.75 = 0.45 \end{aligned}$
::1) P(D和R)=P(D)P(R)=0.60_0.75=0.45

2) $\begin{aligned} P (R a n d D^{C}) = P (R) \cdot P (D^{C}) = 0.75 \cdot 0.40 = 0.30 \end{aligned}$
::2) P(R和DC)=P(R)P(DC)=0.750.40=0.30

3) $\begin{aligned} P (D^{C} a n d R^{C}) = P (D^{C}) \cdot P (R^{C}) = 0.40 \cdot 0.25 = 0.10 \end{aligned}$
::3P(DC和RC)=P(DC)_P(RC)=0.40_0.25=0.10。

Example 3
::例3

Recall the problem from the Introduction: If someone wins the lottery on average once every 100 days, what is the probability that someone will win the lottery in the next 100 days?
::回顾导言中的问题:如果某人每100天平均中彩票一次,那么今后100天中彩票的概率有多大?

Solution:
::解决方案 :

Since there are 100 days and each day has a probability of 0.01 for a lottery winner , then by this logic, there is a 100% chance that a lottery winner in the next 100 days . However, this isn't true because if on average there is a lottery winner once every 100 days, some stretches of 100 days there will be more winners, and some stretches there will be no winners . The 100% solution does not hold.
::因为有100天,每天的彩票赢家的概率是0.01, 那么按照这个逻辑, 在未来100天中,彩票赢家的概率是100%。但是,这不是真的, 因为如果平均每100天中有一个彩票赢家, 平均100天中会有100天, 一定的100天中会有更多的赢家, 而有些则不会有赢家。 100%的解决方案是站不住脚的。

To solve this problem, you need to rephrase the question and ask a slightly different one that will help as an intermediate step. What is the probability that there is not a lottery winner in the next 100 days?
::要解决这个问题, 您需要重新定义问题, 并询问一个稍有不同且能帮助中间一步的版本。在未来100天里没有彩票赢家的可能性有多大 ?

For this to happen, no one can win on day 1, and not win on day 2, and not win on day 3, etc.
::为了实现这一点,没有人能在第1天获胜,没有人在第2天获胜,没有在第3天获胜,等等。

The probability of not winning the lottery on any day is $\begin{aligned} P (n o w i n) = 1 - P (w i n) = 1 - 0.01 = 0.99 \end{aligned}$ .
::在任何一天中不中彩票的可能性是P(无赢)=1-P(赢)=1-0.01=0.99。

The product of each of these probabilities for the 100 days is
::每一100天的这些概率的产物为:

$\begin{aligned} {0.99}^{100} \approx 0.366. \end{aligned}$

Therefore, the probability that there is no lottery winner in the next 100 days is about 36.6%. To answer the original question, the probability that no one wins the lottery in the next 100 days is $\begin{aligned} 1 - 0.366 = 0.634 \end{aligned}$ , or about $\begin{aligned} 63.4 % \end{aligned}$ .
::因此,未来100天没有彩票赢家的概率约为36.6%。要回答最初的问题,未来100天没有人中彩票的概率为1-0.366=0.634,或63.4%左右。

Example 4
::例4

Jack is a basketball player with a free-throw average of 0.77. I f the outcome of a particular free throw is independent of the outcomes of others, w hat is the probability that in a game where he has 8 shots, he makes all 8? What is the probability that he makes only 1?
::杰克是一个篮球运动员,平均自由发射0.77,如果一个特定的自由投球的结果与其他人的结果无关,那么在他有8发球的比赛中,他投出所有8球的概率是多少?他投出1球的概率是多少?

Solution:
::解决方案 :

Let $\begin{aligned} J \end{aligned}$ represent the event that Jack makes the free-throw shot, and $\begin{aligned} J^{C} \end{aligned}$ represent the event that Jack misses the shot.
::让J代表杰克让自由投篮射击的事件, 而JC代表杰克错过射击的事件。

$\begin{aligned} P (J) = 0.77, P (J^{C}) = 0.23 \end{aligned}$

::P(J)=0.77,P(JC)=0.23

The probability that Jack makes all 8 shots is the same as Jack making 1 shot, and making the 2nd shot, and making the 3rd shot, etc.
::杰克射出所有八枪的概率和杰克射出一枪和第二枪的概率一样, 还有第三枪的概率,等等。

$\begin{aligned} P (J)^{8} = {0.77}^{8} \approx 12.36 % \end{aligned}$

::P(J)8=0.778=12.36%

There are 8 ways that Jack could make 1 shot and miss the rest. The probability of each of these cases occurring is
::Jack有八种方法可以打一枪错过其余的

$\begin{aligned} P (J^{C})^{7} \cdot P (J) = {0.23}^{7} \cdot 0.77. \end{aligned}$

::P(JC)7P(J)=0.2370.77。

Therefore, the overall probability of Jack making 1 shot and missing the rest is
::因此,杰克打一枪失去其余的概率是

$\begin{aligned} {0.23}^{7} \cdot 0.77 \cdot 8 = 0.0002097 = 0.02097 % . \end{aligned}$

Example 5
::例5

If it has a 20% chance of raining on Tuesday, your phone has 30% chance of running out of batteries, and there is a 10% chance that you forget your wallet. If the outcome of these particular events are independent, what is the probability that you are in the rain without money or a phone?
::如果星期二有20%的下雨机会,你的手机有30%的电池用完的可能性,还有10%的机会忘记你的钱包。如果这些特定事件的结果是独立的,那么没有钱也没有电话,你在雨中的可能性是多少?

Solution:
::解决方案 :

While a pessimist may believe that all the improbable negative events will occur at the same time, the actual probability of this happening is less than one percent:
::虽然悲观主义者可能认为所有不可能发生的负面事件都会同时发生, 但实际发生的可能性不到1%:

$\begin{aligned} 0.20 \cdot 0.30 \cdot 0.1 = 0.006 = 0.6 % . \end{aligned}$

Example 6
::例6

Consider the previous question with the rain, wallet, and phone. What is the probability that at least one of the three events does occur?
::以雨、钱包和电话来考虑前一个问题。三个事件中至少有一个发生的可能性有多大?

Solution:
::解决方案 :

We can best find this solution by exploring: What is the probability that none of the events occur?
::我们最好通过探索找到这一解决办法:没有发生这些事件的可能性有多大?

$\begin{aligned} 0.8 \cdot 0.7 \cdot 0.9 = 0.504 \end{aligned}$

The probability that at least one occurs is the complement of none occurring.
::至少发生一次的概率是无发生次数的补充。

$\begin{aligned} 1 - 0.504 = 0.496 = 49.6 % \end{aligned}$

Summary
::摘要

The probability of an event is the number of outcomes you are looking for (called successes) divided by the total number of outcomes.
::事件的概率是您正在寻找的结果数量(所谓的成功)除以结果的总数。

The notation $\begin{aligned} P (E) \end{aligned}$ is read "the probability of event $\begin{aligned} E \end{aligned}$ ."
::编号P(E)为“事件E的概率”。

$\begin{aligned} P (E) = \frac{# successful outcomes}{# possible outcomes} \end{aligned}$

::P(E) 成功结果# 可能结果

::标注P(E)为“事件E的概率”。 P(E) 成功结果# 可能的结果#

The complement of an event is the event not happening.
::事件的补充是事件没有发生。

Independent events are events where the occurrence of the 1st event does not impact the probability of the 2nd event.
::独立事件是指第一次事件的发生不影响第二次事件的概率的事件。

Dependent events are events where the occurrence of the 1st event does impact the probability of the 2nd event.
::依附事件是指第一个事件的发生确实影响第二个事件的概率的事件。

Review
::回顾

A card is chosen from a standard deck.
::从标准甲板中选择一张牌。

1. What's the probability that the card is a queen?
::1. 该卡片是女王的概率是多少?

2. What's the probability that the card is a queen or a spade?
::2. 该卡片是皇后还是的概率是多少?

You toss a nickel, a penny, and a dime.
::你扔一个硬币,一个硬币,一个硬币,一个硬币。

3. List all the possible outcomes.
::3. 列出所有可能的结果。

4. What is the probability that the nickel comes up heads?
::4. 镍浮出水面的概率有多大?

5. What is the probability that none of the coins comes up heads?
::5. 没有一枚硬币抬头的概率有多大?

6. What is the probability that at least one of the coins comes up heads?
::6. 至少有一枚硬币浮现出来的可能性有多大?

A bag contains 7 red marbles, 9 blue marbles, and 10 green marbles. You reach into the bag and choose 4 marbles, one after another, without replacement.
::包里装有7个红色大理石、9个蓝色大理石和10个绿色大理石。你伸到包里,一个接一个地选择4个大理石,没有替换。

7. What is the probability that all 4 marbles are red?
::7. 所有4颗大理石都红色的可能性有多大?

8. What is the probability that you get a red marble, then a blue marble, then 2 green marbles?
::8. 获得红大理石、蓝大理石、两颗绿大理石的可能性有多大?

You take a 40-question multiple choice test, and believe that for each question, you have a 55% chance of getting it right.
::你接受一个40个问题多重选择测试, 并相信每个问题, 你有一个55%的机会得到正确的答案。

9. What is the probability that you get all the questions right?
::9. 你答对所有问题的可能性有多大?

10. What is the probability that you get all the questions wrong?
::10. 你把所有问题都弄错的可能性有多大?

A player rolls a pair of standard dice. Find each probability.
::玩家滚动一对标准骰子。查找每个概率。

11. $\begin{aligned} P (s u m i s e v e n) \end{aligned}$
::11. P(总和相等)

12. $\begin{aligned} P (s u m i s 7) \end{aligned}$
::12.(总和为7)

13. $\begin{aligned} P (s u m i s a t l e a s t 3) \end{aligned}$
::13. (总和至少为3)

14. You want to construct a 3-digit number at random from the digits 4, 6, 8, 9 without repeating digits. What is the probability that you construct the number 684?
::14. 您想要从数字4、6、8、9中随机构建一个3位数数字,而不重复数字。您构建数字684的概率是多少?

15. In poker, a straight is 5 cards in a row (for example, 3, 4, 5, 6, 7), NOT all the same suit. (If they are all the same suit, it is considered a straight flush or a royal flush.) A straight can start or end with an ace. What's the probability of a straight? For an even bigger challenge, see if you can calculate the probabilities for all the poker hands.

::15. 在扑克牌中,直牌是5张牌一排(例如3、4、5、6、7),不是全部相同的西装。 (如果它们都是相同的西装,则视为直冲或王牌冲。 )直牌可以用王牌开始或结束。直牌的概率是多少?对于更大的挑战,请看能否计算出所有扑克手的概率。

Review (Answers )
::回顾(答复)

Please see the Appendix.
::请参看附录。

章节大纲

常规

基本概率

Introduction ::导言

Probability ::概率

Examples ::实例

Summary ::摘要

Review ::回顾

Review (Answers ) ::回顾(答复)

Introduction
::导言

Probability
::概率

Examples
::实例

Summary
::摘要

Review
::回顾

Review (Answers )
::回顾(答复)