14.5 预期价值和收益

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• 选择节预期价值和效益

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  预期价值和效益

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  Introduction
  ::导言

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  Suppose you record a pproximately how long it takes you to drive to work each day.
  ::假设你记录你每天开车上班要花多久时间

  Time	Number of Days
  5-7 minutes	1
  7-8 minutes	4
  8-9 minutes	7
  9-10 minutes	9
  10-12 minutes	2

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  How long does it take you to drive to work on average? To answer this question, you could find the weighted average of the expected value. If you choose expected value, consider that the situation gives you frequency rather than probability . You can calculate the probability of each of the categories by dividing each frequency by the total number of days (23). Since the time occurs in intervals, it is reasonable to use the average time in each interval as representative of the category when calculating the expected value.
  ::驱动平均工作需要多长时间? 回答这个问题, 您可以找到预期值的加权平均值。 如果您选择了预期值, 请考虑该情形会给您频率而不是概率。 您可以将每个频率除以总天数( 23) 来计算每个类别的概率。 由于时间间隔间隔不同, 因此在计算预期值时使用每个间隔的平均时间代表该类别的平均时间是合理的 。

  Time	Number of Days	Probability
  5-7 minutes	1	$\begin{array}{r}\frac{1}{23}\end{array}$
  7-8 minutes	4	$\begin{array}{r}\frac{4}{23}\end{array}$
  8-9 minutes	7	$\begin{array}{r}\frac{7}{23}\end{array}$
  9-10 minutes	9	$\begin{array}{r}\frac{9}{23}\end{array}$
  10-12 minutes	2	$\begin{array}{r}\frac{2}{23}\end{array}$

  $\begin{array}{r}6\cdot \frac{1}{23}+7.5\cdot \frac{4}{23}+8.5\cdot \frac{7}{23}+9.5\cdot \frac{9}{23}+11\cdot \frac{2}{23}=\frac{203}{23}\approx 8.8\end{array}$

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  On average, it takes you 8-9 minutes to get to school.
  ::平均来说,你需要8 -9分钟才能上学。

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  When playing a game of chance, there are three basic elements. There is the cost to play the game (usually), the probability of winning the game, and the amount you receive if you win. If games of chance with these three elements are played repeatedly, you can use probability and averages to calculate how much you can expect to win or lose in the long run.
  ::当玩一个机会游戏时,有三个基本元素。玩游戏的成本(通常是),赢得游戏的概率,如果赢的话,你得到的金额。如果与这三个元素的赌博反复玩,你就可以使用概率和平均值来计算从长远看你预期会赢还是输。

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  Consider a dice  game that pays you triple your bet if you roll a 6, and double your bet if you roll a 5. If you roll anything else, you lose your bet. What is your expected return on a $1 wager?
  ::考虑一个骰子游戏,赌注三倍,如果你赌注6,赌注两倍,如果你赌注5,赌注两倍,如果你赌注5,如果你赌注其他什么,赌注就会输掉。1美元赌注的预期回报是什么?

  

   

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  Expected Value
  ::预期值

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  A weighted average is like a regular average, except the data are often given to you in summary form.
  ::加权平均数与正常平均数相同,但数据通常以摘要形式提供给你。

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  Data in Raw Form:
  ::Raw 表单中的数据 :

  1, 3, 5, 3, 2, 1, 2, 5, 6, 4, 5, 2, 6, 1, 4, 3, 6, 1, 2, 4, 6, 1, 3, 1, 3, 5, 6

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  Data in Summary Form:
  ::简表数据:

  Number	Occurrence Count
  1	6
  2	4
  3	5
  4	3
  5	4
  6	5
  Total Occurrences:	27

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  Notice that the summary data indicates, for example, how many times a 1 was rolled (6 times). To calculate the total number of occurrences of data:
  ::请注意,简要数据显示,例如,滚动的数据是每1次(6次)的多少倍。要计算数据发生的总数:

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  • In raw form: Count how many data points you have.
    ::原始形式: 计算您有多少数据点 。
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  • In summary form: Find the sum of the occurrence column.
    ::摘要形式: 查找发生事件栏的总和。

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  To calculate the average:
  ::要计算平均值:

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  • In raw form: Find the sum of the data points, and divide by the total number of occurrences.
    ::原始形式:查找数据点的总和,除以发生次数总数。
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  • In summary form: Find the sum of the data points by finding the sum of the product of each number and its occurrence:
    ::简要形式:通过查找每个数字及其产生结果的总和来查找数据点的总和:

  $$\begin{array}{r}1\cdot 6+2\cdot 4+3\cdot 5+4\cdot 3+5\cdot 4+6\cdot 5=91\end{array}$$

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  Then, divide that sum by the total number of occurrences. In a sense, you are assigning a weight to each of the 6 numbers based on their frequency in your 27 trials. 
  ::然后,将该总数除以发生事件的总数。从某种意义上说,你根据在27次试验中的频率,对6个数字中的每个数字进行加权。

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  The same logic of finding the average of data given in summary form applies when doing theoretical expected value for a game or a weighted average. Consider a game of chance with four prizes ($1, $2, $3, and $4), where each outcome has a specific probability of happening, shown in the table below: 
  ::找到以摘要形式提供的数据的平均值的逻辑在为游戏或加权平均进行理论预期值时同样适用。 考虑一个有四个奖项(1、2、3和4美元)的机会游戏,每个结果都有具体发生概率,如下表所示:

  Number	Probability
  $1	50%
  $2	20%
  $3	20%
  $4	10%

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  Note that the probabilities must add up to 100%!   To calculate the expected value of this game, weight the outcomes by their assigned probabilities.
  ::请注意, 概率必须累积到100% 。 要计算此游戏的预期值, 请按所分配的概率来权衡结果 。

  $$\begin{array}{r}\mathrm{\$}1\cdot 0.50+\mathrm{\$}2\cdot 0.20+\mathrm{\$}3\cdot 0.20+\mathrm{\$}4\cdot 0.10=\mathrm{\$}1.90\end{array}$$

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  This means that if you were to play this game many times, your average amount of winnings should be $1.90. Note there will be no game that you actually get $1.90, because that was none of the options. Expected value is a measure of what you should expect to get per game in the long run.
  ::这意味着,如果你玩这个游戏很多次, 你的平均获奖额应该是1. 90美元。 注意, 不会有任何游戏, 你实际上可以得到1. 90美元, 因为没有这个选项。 期望值是衡量您期待每个游戏长期获得多少的尺度 。

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  Play, Learn, and Explore with Expected Value: 
  ::以预期值播放、学习和探索:

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  The payoff of a game is the expected value of the game minus the cost. If you expect to win about $1.90 on average when you play a carnival game repeatedly, and it costs only $1.50 to play, then the expected payoff is 40 cents per game.
  ::游戏的回报是游戏的预期值减去成本。 如果您期望在反复玩嘉年华游戏时平均赢1. 90美元, 而游戏只花费1. 50美元, 那么每场比赛的预期回报是40美分。

  

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  In general, to find the expected value for a game or other scenario, find the sum of all possible outcomes, each multiplied by the probability of its occurrence.
  ::一般而言,为了找到一种游戏或其他情景的预期值,找到所有可能结果的总和,每个结果乘以其发生概率。

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  The following video  defines expected value, and provides two examples of how to find the expected value of an event : 
  ::以下视频界定了预期价值,并提供了两个实例,说明如何找到某一事件的预期价值:

   

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  Play, Learn, and Explore with Expected Value and Payoff: 
  ::以预期值和收益来玩、学习和探索:

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  Examples
  ::实例

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  Example 1
  ::例1

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  A teacher has five categories of grades that each make up a specific percentage of the final grade. Calculate Owen's grade.
  ::教师有五类等级,各占最后年级的具体百分比。计算Owen的等级。

  Category	Weight	Owen's grade
  Quizzes and Tests	30%	78%
  Homework	25%	100%
  Final	20%	74%
  Projects	20%	90%
  Participation	5%	100%

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  Solution:
  ::解决方案 :

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  Using the concept of weighted average, weight each of Owen's grades by the weight of the category.
  

  $$\begin{array}{r}0.78\cdot 0.3+1\cdot 0.25+0.74\cdot 0.20+0.90\cdot 0.20+1\cdot 0.05=0.862\end{array}$$

  
  ::使用加权平均数概念,欧文各等级的加权加权平均数按该类别的加权权重计算。 0.78_0.3+1}0.25+0.74_0.20+0.0.0.0.0.0.20+0.0.0.0.0.20+10.05=0.862

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  Owen gets an 86.2%.
  ::Owen拿到86.2%

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  Example 2
  ::例2

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  Courtney plays a game where she flips a coin. If the coin comes up heads, she wins $2. If the coin comes up tails, she loses $3. What is Courtney's expected payoff each game?
  ::Courtney在玩游戏,她把硬币翻了个硬币。如果硬币浮出头来,她就会赢2美元。如果硬币翻了尾巴,她就会输3美元。Courtney每场比赛的预期回报是什么?

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  Solution:
  ::解决方案 :

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  The probability of getting heads is 50%, and the probability of getting tails is 50%. Using the concept of weighted averages, you should weight winning $2 and losing $3 dollars by 50% each. In this case, there is no initial cost to the game.
  ::获得头部的概率是50%,获得尾部的概率是50%。使用加权平均值的概念,你应该体重在2美元中得分,每损失3美元,每损失50%。在这种情况下,游戏没有初始成本。

  $$\begin{array}{r}2\cdot 0.50-3\cdot 0.50=-0.50\end{array}$$

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  This means that while sometimes she might win and sometimes she might lose, on average she is expected to lose about 50 cents per game.
  ::这意味着,虽然有时她可能会赢,有时她可能会输,但平均而言,她每场比赛都会损失大约50美分。

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  Example 3
  ::例3

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  Paul is deciding whether or not to pay the parking meter when he is going to the movies. He knows that a parking ticket costs $30, and he estimates there is a 40% chance that the traffic police spot his car and write him a ticket. If he chooses to pay the meter, it will cost $4, and he'll have a 0% chance of getting a ticket. 
  ::保罗正在决定是否在去看电影时支付停车表。他知道停车罚单需要30美元,他估计交通警察发现他的车并给他写一张罚单的可能性为40%。 如果他选择支付停车罚单,他将花费4美元,而他获得停车罚单的机会为零。

  

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  Is it cheaper to pay the meter or risk the fine?
  ::支付计时表还是冒罚款风险更便宜吗?

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  Solution:
  ::解决方案 :

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  Since there are two possible scenarios, calculate the expected cost in each case.
  ::由于有两种可能的设想,计算每种情况下的预期费用。

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  $$\begin{array}{r}Payingthemeter:\mathrm{\$}4\cdot 100\mathrm{\%}=\mathrm{\$}4\end{array}$$

  
  ::支付计时费:4100美元 4美元

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  $$\begin{array}{r}Riskingthefine:\mathrm{\$}0\cdot 60\mathrm{\%}+\mathrm{\$}30\cdot 40\mathrm{\%}=\mathrm{\$}12\end{array}$$

  
  ::面临罚款风险:0.60美元 30美元 40美元 12美元

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  Risking the fine has an expected cost three times that of paying the meter.
  ::风险罚款的预期费用是支付计时费的三倍。

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  Example 4
  ::例4

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  Recall the problem  from the Introduction: Suppose you are playing a dice game that pays you triple your bet if you roll a 6, and double your bet if you roll a 5. If you roll anything else, you lose your bet. What is your expected return on a $1 wager? 
  ::回顾引言中的问题:假设你玩骰子游戏,赌注赌注三倍,赌注两倍,赌注两倍,赌注两倍,赌注5,赌注两倍,赌注两倍,赌注两倍,赌注两倍,赌注两倍,赌注两倍,赌注两倍,赌注一半,赌注一半,赌注一半,赌注一半,赌注一半,赌注多少?

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  Solution:
  ::解决方案 :

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  In a game that pays you triple your bet if you roll a 6, and double your bet if you roll a 5, the expected return on a $1 wager is
  ::赌注是赌注的三倍 赌注是赌注的两倍 赌注是赌注的两倍 赌注是赌注的五倍 赌注是赌注的1美元

  $$\begin{array}{r}\mathrm{\$}0\cdot \frac{2}{3}+\mathrm{\$}2\cdot \frac{1}{6}+\mathrm{\$}3\cdot \frac{1}{6}=\frac{5}{6}.\end{array}$$

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  If you spend $1 to play the game, and you play the game multiple times, you can expect a return of   $\begin{array}{r}\frac{5}{6}\end{array}$ of $1, or about 83 cents on average.
  ::如果你花一美元玩游戏, 玩游戏多次, 你可以指望回报56美元, 平均约83美分。

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  Example 5
  ::例5

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  What is the payoff of a slot machine that costs $1 to play and pays out $5 with probability 4%, $10 with probability of 2%, and $30 with probability 0.5%?
  ::一台要花一美元才能玩的机房机器的回报是什么? 机房机要花5美元,概率为4%,10美元,概率为2%,30美元,概率为0.5%。

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  Solution:
  ::解决方案 :

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  $$\begin{array}{r}0\cdot 0.935+5\cdot 0.04+10\cdot 0.02+30\cdot 0.005-1=-0.45\end{array}$$

  You will lose 45 cents on average if you play the slot machine many times.
  ::00.935+50.04+100.02+300.005-10.45 如果你玩多次时空机,平均会损失45美分。

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  Example 6
  ::例6

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  What is the expected value of an experiment with the following outcomes and corresponding probabilities?
  ::对以下结果和相应概率进行试验的预期价值是什么?

  Outcome	31	35	37	39	43	47	49
  Probability	0.1	0.1	0.1	0.2	0.2	0.2	0.1

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  Solution:
  ::解决方案 :

  $$\begin{array}{r}31\cdot 0.1+35\cdot 0.1+37\cdot 0.1+39\cdot 0.2+43\cdot 0.2+47\cdot 0.2+49\cdot 0.1=41\end{array}$$

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  Summary
  ::摘要

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  • A weighted average is an average that multiplies each component by a factor representing its frequency or probability. 
    ::加权平均数是每个组成部分乘以代表其频率或概率的一个系数的平均数。
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  • The expected value is the return or cost you can expect on average, given many trials.
    ::预计价值是,鉴于许多审判,平均回报率或预期成本。
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  • The payoff of a game is the expected value of the game minus the cost.
    ::游戏的回报是游戏的预期值减去成本。

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  Review
  ::回顾

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  1. Explain how to calculate expected value.
  ::1. 解释如何计算预期值。

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  2. True or false: If the expected value of a game is 50 cents, then you can expect to win 50 cents each time you play.
  ::2. 真实的或虚假的:如果游戏的预期值是50美分,那么每玩一次,你就可以指望赢得50美分。

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  3. True or false: The greater the number of games played, the closer the average winnings will be to the theoretical expected value.
  ::3. 真实或虚假:游戏次数越多,平均赢得越接近理论预期值。

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  4. A player rolls a standard pair of dice. If the sum of the numbers is a 6, the player wins $6. If the sum of the numbers is anything else, the player has to pay $1. What is the expected value for this game?
  ::4. 玩家滚动一对标准骰子。如果数字的总和是6,玩家将赢得6美元。如果数字的总和是其它东西,玩家必须支付1美元。本游戏的预期值是多少?

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  5. What is the payoff of a slot machine that costs 25 cents to play and pays out $1 with probability 10%, $50 with probability of 1%, and $100 with probability 0.01%?
  ::5. 一台要花25美分才能玩的空档机的回报是什么? 机票1美元,概率为10%,概率为50美元,概率为1%,概率为100美元,概率为0.01%。

   

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  6. A slot machine pays out $1 with probability 5%, $100 with probability of 0.5%, and $1,000 with probability 0.01%. If the casino wants to guarantee that it won't lose money on this machine, how much should it charge people to play?
  ::6. 空位机支付1美元,概率为5%,100美元,概率为0.5%,1 000美元,概率为0.01%。 如果赌场想保证它不会在机器上损失钱,那么它应该给人们多少打球费?

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  7. What is the expected value of an experiment with the following outcomes and corresponding probabilities?
  ::7. 对以下结果和相应概率进行试验的预期价值是什么?

  Outcome	12	14	18	20	21	22	23
  Probability	0.05	0.1	0.6	0.1	0.1	0.03	0.02

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  Calculate the final grades for each of the students given the information in the table.
  ::计算表格中信息显示的每个学生的最后年级。

  Category	Weight	Sarah	Jason	Kimmy	Maria	Kayla
  Quizzes and Tests	30%	74%	85%	90%	80%	75%
  Homework	25%	95%	40%	100%	90%	95%
  Final	20%	68%	80%	85%	70%	50%
  Projects	20%	85%	70%	95%	75%	85%
  Participation	5%	95%	100%	100%	80%	60%

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  8. What is Sarah's final grade?
  ::8. Sarah的最后一年级是多少?

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  9. What is Jason's final grade?
  ::9. 杰森的最后一年级是多少?

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  10. What is Kimmy's final grade?
  ::Kimmy的最后一年级是多少?

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  11. What is Maria's final grade?
  ::11. 玛丽亚的最后一年级是多少?

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  12. What is Kayla's final grade?
  ::12. Kayla的最后一年级是多少?

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  13. Look back at the grades and final grades for the five students. Do the grades seem fair to you given how each student performed in each area? Do you think the category weights should be changed?
  ::13. 回顾五个学生的年级和最后年级,考虑到每个学生在各领域的表现,这些年级是否公平?你认为分类权重应该改变吗?

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  14. You are in charge of a game booth at the fair. In the game, players pick a card at random from the deck. If the card is a jack, queen, or king, the player wins $5. What is the minimum amount you should charge to feel confident you will make a profit by the end of the fair?
  ::14. 你负责博览会的一个游戏摊位,在比赛中,球员随机从甲板上挑一张牌,如果牌牌是J、Q或K,球员赢5美元。

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  15. Make up your own game that has at least two possible outcomes with an expected payoff of 50 cents.
  ::15. 形成自己的游戏,至少产生两个可能的结果,预期回报为50美分。

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  16. Explain why it makes sense for a casino to consider the concept of expected value when designing its games.
  ::16. 解释为什么赌场在设计其游戏时考虑预期价值的概念是有道理的。

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  Review (Answers)
  ::回顾(答复)

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  Please see the Appendix.
  ::请参看附录。