15.2 预期价值和收益
Section outline
-
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)、赢游戏的概率和赢球后你得到的金额。如果与这三个元素的赌博游戏反复玩,您可以使用概率和平均值来计算您在长期内希望赢还是输。Consider a dice game that pays you triple your bet if you roll a six and double your bet if you roll a five. If you roll anything else you lose your bet. What is your expected return on a one dollar wager?
::考虑一个骰子游戏,赌注三倍,如果你滚六,赌注两倍,如果你滚五,赌注两倍。如果你滚其他什么,赌注就会输掉。你一美元赌注的预期回报是什么?Expected Value and Payoffs
::预期价值和效益There are two ways to be given data, raw form and summary form. The following data represents which numbers are rolled with a standard six-sided dice:
::提供数据有两种方式,即原始形式和摘要形式。以下数据表示哪些数字以标准的六面骰子滚动: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
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 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:-
In raw form: count how many data points you have
::原始形式: 计算您有多少数据点 -
In summary form: find the sum the occurrence column
::摘要形式: 查找发生总和
To calculate the average:
::要计算平均值:-
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,除以发生次数的总数。 -
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:通过找到每个数字及其产生结果的总和,找到数据点的总和:
Then, divide that sum by the total number of occurrences. In a sense, you are assigning a weight to each of the six numbers based on their frequency in your 27 trials.
::然后,将这个总数除以发生事件的总数。从某种意义上说,你根据在27次试验中的频率对六个数字中的每一个数字进行加权。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. The expected value is the return or cost you can expect on average, given many trials. A weighted average is an average that multiplies each component by a factor representing its frequency or probability. A weighted average is like a regular average except the data is often given to you in summary form.
::在对游戏或加权平均进行理论预期值或加权平均值时,也适用找到以摘要形式提供的数据的平均值的逻辑。预期值是,考虑到多次试验,平均回报率或成本是您可以预期的平均回报率或成本。加权平均值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值。加权平均值与正常平均数一样,但通常以摘要形式提供的数据除外。Consider a game of chance with 4 prizes ($1, $2, $3, and $4) where each outcome has a specific probability of happening, shown in the table below:
::如下表所示,在每种结果都有具体发生可能性的4个奖项(1、2、3和4)的赌博中,每个结果都有具体发生的可能性:Number Probability $1 50% $2 20% $3 20% $4 10% Note that the probabilities must add up to 100%. In order to calculate the expected value of this game, weight the outcomes by their assigned probabilities.
::请注意概率必须达到100%。 为了计算此游戏的预期值, 请按分配给它们的概率来权衡结果 。This means that if you were to play this game many times, your average amount of winnings should be $2.20. Note that there will be no game that you actually get $2.20, 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2. 20美元。 请注意, 不会有任何游戏, 你实际上可以得到220美元, 因为没有这个选项。 期望值是衡量每个游戏从长远看应该得到多少的尺度 。The payoff of a game is the expected value of the game minus the cost. If you expect to win about $2.20 on average if you play a game repeatedly and it costs only $2 to play, then the expected payoff is $0.20 per game.
::游戏的回报是游戏的预期值减去成本。 如果您希望赢得平均2. 20美元, 如果你再玩一次游戏,而且游戏只花2美元, 那么每场比赛的预期回报是2.0美元。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。Examples
::实例Example 1
::例1Earlier, you were asked to consider a dice game that pays you triple your bet if you roll a six and double your bet if you roll a five. For this game, the expected return on a one dollar wager is:
::早些时候,有人要求你考虑一个骰子游戏,如果赌6次,赌3倍。如果赌5次,赌2倍。对于这个游戏,一美元赌注的预期回报是:If you spend $1 to play the game and you play the game multiple times, you can expect a return of of one dollar or about 83 cents on average.
::如果你花一美元玩游戏, 玩了多次游戏, 你可以指望回报56美元, 平均约83美分。Example 2
::例2What 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 Example 3
::例3A 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。计算欧文的等级。Category Weight Owen’s grade Quizzes and Tests 30% 78% Homework 25% 100% Final 20% 74% Projects 20% 90% Participation 5% 100% Using the concept of weighted average, weight each of Owen’s grades by the weight of the category.
::使用加权平均数概念,欧文的每个职等按类别加权加权加权。Owen gets an 86.2%.
::Owen拿到86.2%Example 4
::例4Courtney 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玩游戏,她把硬币抛在了一起。如果硬币浮上头,她就会赢2美元。如果硬币浮上尾巴,她就会输3美元。 Courtney每场比赛的预期回报是什么?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 dollars 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,你应该体重赢得2美元,每损失3美元50%。在这种情况下,游戏没有初始成本。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。Example 5
::例5Paul 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 that 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 dollars and he will have a 0% chance of getting a ticket.
::保罗正在决定他去看电影时是否要支付停车表。 他知道停车罚单需要30美元,他估计交通警察有40%的机会发现他的车,并给他写一张票。 如果他选择支付停车表,他将花费4美元,他将获得车票的可能性为零。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?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。
::支付计时费:4100美元 4美元
::面临罚款风险:0.60美元 30美元 40美元 12美元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。Summary -
Expected value
is the return or cost you can expect on average, given many trials.
::预计值是考虑到许多试验,你平均预期的回报率或成本。 -
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。 -
To calculate the expected value, weigh the outcomes by their assigned probabilities and find the sum of all possible outcomes, each multiplied by the probability of its occurrence.
::为计算预期值,按所分配的概率权衡结果,并找出所有可能结果的总和,每乘其发生概率。 -
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。
Review
::回顾1. Explain how to calculate expected value.
::1. 解释如何计算预期值。2. True or false: If the expected value of a game is $0.50, then you can expect to win $0.50 each time you play.
::2. 真实的或虚假的:如果游戏的预期值为0.50美元,那么每次玩游戏,你就可以指望赢得0.50美元。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。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?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%。6. A slot machine pays out $1 with probability 5%, $100 with probability of 0.5%, and $1000 with probability 0.01%? If the casino wants to guarantee that they won’t lose money on this machine, how much should they 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%? 如果赌场想保证他们不会在机器上输钱,他们应该收取多少钱?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 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 Kimy 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% 8. What is Sarah’s final grade?
::8. Sarah的最后一年级是多少?9. What is Jason’s final grade?
::9. 杰森的最后一年级是多少?10. What is Kimy’s final grade?
::10. Kimy的最后一年级是多少?11. What is Maria’s final grade?
::11. 玛丽亚的最后一年级是多少?12. What is Kayla’s final grade?
::12. Kayla的最后一年级是多少?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 of the areas? Do you think the category weights should be changed?
::13. 回顾这五名学生的年级和期末成绩,考虑到每个学生在每一领域的表现,这些年级是否公平?你认为分类的权重应该改变吗?14. You are in charge of a booth for a game at the fair. In the game, players pick a card at random from the deck. If the card is a J, Q, or K, the player wins $5. What is the minimum amount you should charge in order to feel confident you will make a profit by the end of the fair?
::14. 你负责集市游戏的摊位,在比赛中,玩家随机从甲板上选一张牌,如果牌是J、Q或K,玩家赢5美元。你应该收取多少最低金额才能有信心在集市结束时获利?15. Make up your own game that has at least 2 possible outcomes with an expected payoff of $0.50.
::15. 形成自己的游戏,至少产生两个可能的结果,预期收益为0.50美元。16. Explain why it makes sense for a casino to consider the concept of expected value when designing their 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。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 。 -
In raw form: count how many data points you have