模拟模拟

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Lesson Objectives
::经验教训目标

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• Decide if a specified model is consistent with results using simulation.
  ::决定特定模型是否与使用模拟的结果一致。
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• Use probabilities to make fair decisions.
  ::利用概率作出公平决定。
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• Analyze decisions and strategies using probability concepts.
  ::利用概率概念分析决定和战略。

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Introduction: A Day At The Carnival
::导言:嘉年华节

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Lorenzo and Cheyenne are at the carnival and are playing the game shown below. Lorenzo thinks that the best way to get 5 tickets is to drop the ball directly above the 5, and Cheyenne thinks that it doesn't matter where you drop the ball, claiming that it will be equally likely to end up in any slot along the bottom. Use the interactive below to drop the ball in the same spot repeatedly and check to see whether some outcomes are more likely to occur than others.
::Lorenzo和Cheyenne在嘉年华会上玩下面的游戏。Lorenzo认为获得5张球票的最好方式是直接投到5号球上方,Cheyenne认为投球的位置并不重要,声称它同样有可能出现在底部的任何位置。用下面的交互式方式反复投到同一个位置,并检查是否比其他位置更可能出现一些结果。

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Discussion Question: Do you agree with Lorenzo or Cheyenne?
::讨论问题:你同意洛伦佐还是夏安?

 

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Activity 1: Simulations
::活动1:模拟

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One way to approach the problem presented in the introduction is to use a simulation. A simulation is the use of a mathematical model to recreate an outcome under specific conditions. Computers allow us to run simulations millions of times, making it easy to identify which outcomes are more likely to happen accurately. We can run the simulation thousands of times to get a better picture of Lorenzo's strategy. Perform a large-scale simulation using the interactive below.
::处理导言中提出的问题的一种方法是使用模拟。模拟是使用数学模型在特定条件下重新创造结果。计算机让我们可以进行数百万次模拟,从而更容易地确定更可能准确实现的结果。我们可以进行数千次模拟,以便更好地了解洛伦佐的战略。使用下面的交互式模拟进行大规模模拟。

 

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Discussion Question: How do you think that dropping the ball from different starting points will affect the probabilities?
::讨论问题:你认为从不同起点投球会如何影响概率?

 

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Activity 2: Beating the Shift
::活动2:击败转变

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Every team in major sports wants to get an edge over their competitors, with many having taken to statistics to develop strategies that will optimize their chances of success. As a result, statisticians are in demand. One example of how statistics can shape the way sports are played is the use of the shift in baseball. A shift is when you move the infielders to one side of the field when you think that a batter is more likely to hit the ball to that side, thus increasing your chances of getting the batter out. However, most shifts leave the other side of the field open, virtually assuring the batter of a hit if he or she can hit the ball to that side of the field.
::参加大型体育赛事的每个球队都想赢得竞争对手的优势,许多球队都接受统计,以制定最佳成功机会的战略。因此,统计人员需要。统计数字如何塑造体育运动方式的一个例子就是棒球的变换。一个转变是,当你认为击球手更可能打球到球到球场一边时,你就会把球推到球的一边,从而增加你把球打出来的机会。然而,大多数轮班都让球场的另一边开着门,几乎保证球手会打到球到球场的那一边。

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Terry wants to determine if the infield shift is effective at lowering a batter's success rate in reaching base on a ball put in play. He took a sample of 100 balls put into play against the shift. Of those balls put into play, 25 were categorized as successful, resulting in the player reaching base. Since 2000, the average major league success rate of reaching base on a ball put in play with no shift is around 30%, or .300 in baseball terminology. This leads Terry to assume that the shift does lower the success rate of the batter in reaching base.
::泰瑞想确定场内轮班是否有效降低打球手在球场上的成功率。 他选取了100个球在球场上打球的样本。 在这些球中, 25个球被归类为成功, 导致球员进入球场。 自2000年以来, 在球场上打球的平均主要联赛成功率是30%左右, 或者说棒球术语是300。 这导致泰瑞认为球队在球场上打球的成功率会降低。

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The simulation below will allow you to use a simulation when no shift is used to compare to Terry's shift sample. Run a few simulations using the interactive below to help answer the following questions about Terry's sample.
::以下的模拟将允许您在不使用转换来比较 Terry 的变换样本时使用模拟。 使用下面的交互式模拟运行一些模拟, 以帮助回答关于 Terry 样本的以下问题 。

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Discussion Question: Referring to the questions in the question set above, if replacing your goalie with an extra offensive player increases your probability of scoring a goal, why do teams play with a goalie at all?
::讨论问题:提到上述问题中的问题,如果用一个额外的攻击性玩家取代守门员会增加你赢得一个球分的概率,为什么团队会与守门员一起玩?

 

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Activity 3: The Monte Carlo Simulation
::活动3:蒙特卡洛模拟

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The Monte Carlo simulation gets its name from the casino game Monte Carlo R oulette. In this game, players predict whether a number chosen at random between 0 and 36 will be red or black. Out of the 37 numbers, 18 of them are red, and 18 of them are black. Choosing correctly will double the money you bet, and choosing incorrectly will cost you all of the money you bet. The trick is that 0 is neither red nor black; it's green. We can use this information to predict the likelihood of success.
::蒙特卡洛模拟会从赌场游戏蒙特卡洛·鲁莱特获得它的名字。在这个游戏中,玩家预测一个在0到36之间随机选择的数字是红色还是黑色。在37个数字中,18个是红色,18个是黑色。正确选择会加倍你赌注的钱,错误选择会使你赌注的全部钱花光。诀窍是零既不是红色,也不是黑色,而是绿色的。我们可以使用这些信息预测成功的可能性。

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• Probability of winning =  $\begin{array}{r}\frac{18}{37}\approx 0.486\text{ or }48.6\mathrm{\%}\end{array}$  
  ::获胜概率=1837=0.486或48.6%
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• Probability of losing =  $\begin{array}{r}\frac{19}{37}\approx 0.514\text{ or }51.4\mathrm{\%}\end{array}$  
  ::损失概率=1937 = 0.514或51.4%
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• Net probability of success =  $\begin{array}{r}\frac{18}{37}-\frac{19}{37}=\frac{-1}{37}\approx -0.027\text{ or }-2.7\mathrm{\%}\end{array}$  
  ::成功概率净额=1837-1937=137=0.027或-2.7%

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Let's run a simulation of 100 bets in Monte Carlo Roulette. To keep the math simple, every bet will be for $1. The way that the simulation works is that n simulations are run for n bets of $1. Each bet has a 48.6% chance of winning. The simulation will run through n bets and makes a map of all the outcomes. The resulting graph makes it easier to see the patterns that arise. For every dollar you bet, you can expect to make   $\begin{array}{r}\$1\times -0.027=-\$0.027\end{array}$ , or in other words, we expect to lose 2.7 cents.  
::让我们在蒙特卡洛鲁莱特模拟100个赌注。 为了保持数学简单, 每个赌注都将是1美元。 模拟工作的方式是 n 模拟的赌注是1美元。 每个赌注的赢率是48.6%。 模拟将进行n 赌注, 并绘制所有结果的地图。 由此得出的图表可以更容易地看到出现的模式。 每赌一美元, 你就可以指望每赌一美元 0.027 $0.027, 或者换句话说, 我们预期会损失2. 7 美分。

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Monte Carlo simulations are integral in risk assessment for fields like financial planning and money lending. Investors and lenders use Monte Carlo simulations to explore all possible outcomes based on different variables. Below we will explore a simple example of a Monte Carlo Simulation based on . The interactive below displays a colored background where a dart is thrown at random. We can use the Monte Carlo simulation to determine the probability of the dart randomly landing on a color.
::Monte Carlo模拟是财务规划和货币贷款等领域风险评估不可或缺的组成部分。投资者和贷款人利用Monte Carlo模拟根据不同的变量探索所有可能的结果。下面我们将探讨一个基于 .的 Monte Carlo模拟的简单例子。下面的互动展示了随机投射飞镖的彩色背景。我们可以使用 Monte Carlo模拟来确定飞镖随机降落在颜色上的概率。

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At a carnival, a dart is randomly launched at the board below with an equal probability of landing anywhere on the board. You must predict whether the dart will land inside or outside the circle. You will earn 2 tickets if you correctly predict that the dart will land inside the circle and 7 tickets if you correctly predict that the dart will land outside the circle. The game costs $1 to play.
::在狂欢节上, 飞镖在下面的板块上随机发射, 其降落概率在板板上任何地方。 您必须预测飞镖是否会在圆圈内或外降落。 如果您正确预测飞镖将在圆圈内降落, 您将获得两张票。 如果您正确预测飞镖将在圆圈外降落, 将会获得七张票, 如果您正确预测飞镖将在圆圈外着陆, 您将获得两张票。 游戏需要花1美元才能进行游戏 。

 

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Discussion Question: What is the probability of landing a dart inside the circle multiplied by 4? Why?
::讨论问题:在圆内降落飞镖的概率乘以4是多少?为什么?

Summary

播放段落 A simulation is the use of a mathematical model to recreate an outcome under specific conditions. ::模拟是指使用数学模型在特定条件下重新产生结果。 播放段落 Simulations can be used to predict the likelihood of a specific outcome. ::模拟可用于预测具体结果的可能性。

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Wrap-Up: Review
::总结:审查