3.2: Perception Process—Parts 1 and 2 (Selection and Organization)
Basic Components of the Perception Process
The perception process has three stages: selection, organization, and interpretation (Knudsen, et al., 2021). As shown in Figure \(\PageIndex{1}\), the perception process is both cognitive and psychological. It can influence how we communicate with ourselves and others. Look at any number of objects around you. What you choose to focus on, how you would describe it, and finally what you think about that object is unique to you. In this section, we walk you through some of the basic components of the perception process. The goal here is to gain a better understanding of how our perceptions are formed, giving us some unique insights into how those impact our interpersonal relationships.
Selection
The first step in the perception process is selection . To build a foundation for selection, we must first attend to the senses. On any given day, our senses (sight, hearing, touch, taste, and smell) are bombarded with stimuli. Because we cannot respond to them all, we will choose what to pay attention to and when. This process is called sensory selection . Sensory selection is a process for determining which stimulus gets our attention. This often occurs subconsciously and with little effort on our behalf (Knudsen, et al. 2021). So in our very busy lives, how do we choose? Let’s dive deeper to see how the selection process is influenced by salience, needs, interests, and expectations to shape our perceptions.
Salience
Salience is a key component of understanding how and why we pay attention to. Salience is defined as anything that attracts our attention (Perception Process, 2020). Based on context, anything from an object, an idea, a quote, a concept, or a particular person can be salient to us. For example, say you are looking at the Instagram feed of your local news organization. There is a story about the rise in college tuition. As a current college student, you would likely stop and read that article. However, your friend, a recent college graduate, may not even take note of the headline. The post is not inherently more important than anything else on your feed. The importance lies in the salience to you, the viewer. With so much information at our fingertips, our brains will pick and choose, or select, the information we pay attention to and what to ignore. This will directly impact our unique perceptions. Let’s look at some influences that make stimuli salient.
Needs
Our basic needs often drive our perception. If you are hungry, you will mostly see opportunities to eat. You might notice restaurant fronts, hear ads for takeout on the radio, or see a person eating an ice cream cone walking down the street. You are selecting (most likely unconsciously) stimuli related to your basic need of hunger. Now let’s consider needs as they relate to relationships. According to Maslow’s Hierarchy of Needs (Maslow, 1943), humans have a need for love and belongingness. In times when we might feel lonely or isolated, we may be especially in tune to the nonverbal cues of our friends or significant other. Or we might take special note of someone in class who goes above and beyond to give us special attention and care. These cues may not be relevant to those around us, but at times when this need is triggered, we may select to note the actions of others that may otherwise seem mundane.
Let’s consider how we could use our understanding of needs in the selection process to communicate with our loved ones more effectively. Take for example that we are feeling particularly disconnected from a friend. Because our perception of what is happening is based on our own needs at that time, our perception may not match up with our friend's perception of the situation. As an example, imagine a friend in a hurry to get to work, but doing their best to be kind and communicative. This friend may not notice that because they are in a hurry, their communication seems terse or abrupt. Should you communicate to that friend that you are feeling especially lonely or isolated? Communicating our needs and perceptions of the situation with each other is a positive step. If your friend knows you feel unsure or concerned about the relationship, perhaps they will take extra care to check in with you or make an effort to not be so short on replies via text. Effective communication means that instead of jumping to conclusions, we recognize our needs, identify what would help fulfill them, and then use interpersonal communication. This will help to avoid hurt feelings, and instead go towards building maintaining a healthy relationship. Other influences on our perceptions include our interests and expectations.
Let’s take a moment to check in with ourselves as we read this chapter.
How Are You Feeling?
If you are tired, hungry, busy, stressed, or distracted, you may be having a hard time focusing on this chapter. If you are calm, warm, comfortable, and focused, you may have a better time tuning in with our concepts.
Take a moment to scan your senses. Is there anything you could do at this moment to alter your state for a favorable outcome: the ability to enjoy and understand learning about the perception process as you read more?
Expectations
Perhaps one of the most powerful and truly impactful aspects of selection is expectations . Expectations can be seen as a two-sided coin. On one hand, we pay attention to those things we think should happen, and on the other hand, we pay attention to things that violate our expectations.
Let’s look at an example of how we select to pay attention to what should not have happened. Say you get up earlier than normal one day just to send your significant other a quick “I love you, have a great day” message before they head to work. You don’t hear back. Hours go by. When your inbox is still empty, you are on high alert. There was nothing to signal the silence, no fight or argument the day before that would warrant no response, so it is unexpected. You jump on social media and they’re tagged in a video with a friend, laughing and singing. Now you know they are OK, and apparently in good spirits, so this lack of response is even more unexpected. You begin to feel angry, anxious, hurt, and confused. You start to pay attention to things you would have otherwise found mundane: the time the video was posted, who else is around, what your significant other is wearing, etc. You decide to leave an angry comment on the social media post.  You learn in your conversation that they not only left their phone at home in a rush to make it to work on time but weren't expecting to hear from you until after their shift, per your normal routine. You realize that your anger is not necessarily based on objective facts, but on your perception of what your partner should have done.
The trick is learning how to separate our expectations from our reality. Can we objectively measure anything, really? Our perceptions influence our reality, not the other way around. Therefore, the goal is not to void ourselves of past experiences, but to be aware of how our needs, interests, and expectations shape our perceptions.
Organization
Now that we understand how we select which stimuli to pay attention to, let’s look at how we organize that information. How we understand this process of organization comes from gestalt theory . Gestalt is German for “pattern” or “shape,” and the theory asserts that we essentially process stimuli by blending external stimuli with internal processes (Rock & Palmer, 1990). In essence, how we perceive the external world is heavily determined by internal influences. To break this down further, we will look at how we organize stimuli based on the three most common factors: similarity, difference, and proximity (Coren, 1980).
Assumed Similarity and Difference
When we believe or sense that someone is similar to us, we are more likely to be attracted to them as friends and give them the benefit of the doubt. The opposite is also true: if we find someone to be different from us, we will distance ourselves. It would be very hard to start this section off by not blatantly acknowledging that assumed similarities and differences often stem from stereotypes. So let me start with why we might stereotype in the first place: To put it simply, because it is easy. As we mentioned earlier, we are bombarded by stimuli on a daily basis. To be efficient—which many of us know is not always advantageous—we may stereotype because of its appeal to simplicity.
Stereotypes have been described in psychology as “allowing easier and more efficient processing of information” (Hilton & Hippel, 1996, p. 240). Our brains are trained, from early childhood on, to function in this way. Whether it’s learning about animals, shapes, colors, or letters, we are taught to group similar-looking objects together as a means of understanding our environment. Through repetition of such activities, our perceptions based on similarities and differences are formed, reinforced, and encouraged. Although stereotyping can serve as a path of least resistance, we know that it can also create barriers for our ability to create healthy relationships with others. Now that we understand why we may stereotype subconsciously, and how that can be dangerous, let us continue to build these assumed realities.
Proximity
Proximity refers to how we perceive one object based on its surroundings. So, as we go about our day, we do not see objects, situations, or scenarios as isolated, but in context to the surroundings. This concept has been highlighted in the figure-ground relationship, which argues that our focus, whether it be on the object or the background, will change our perception (Knudsen, et al., 2021). This is highlighted in the classic faces/vase image displayed in Figure (\PageIndex{4}\). Depending on whether you focus on the black or white portions of the image, this determines which figure becomes prominent. Try for yourself. Do you see different the vase or the profiles?
Let’s add a layer here, building on our past key terms: salience and selection. Let’s say that you see a couple on a busy commercial street arguing, perhaps screaming. You may look at them and think, "wow, that is strange." You may attribute their behaviors to many factors, but you continue to drive, as you do not know the individuals or the situation. Now, let’s say you turn onto your street, get very close to your home, and see two people arguing, perhaps screaming. This is very close to where you live. These people could be your neighbors. You now feel that, because of the proximity of this confrontation to your home, it is more salient. You may also select to pay close attention to the two people arguing; you may even decide to take action, such as stepping in or calling the police, because you may organize the event as a threat to you and your family’s safety. The situation is the same, but the surroundings of the people in conflict have changed, thus making your perception of the situation different.
Before we leave this section on proximity, let’s acknowledge how commonly we, and others, use proximity to organize the intentions of those around us. Have you ever been out with a friend to have someone assume you were together, or on a date? Or if you encounter the scene in Figure \(\PageIndex{5}\), where one child stands over another, who is on the ground and crying, could there be an assumption that one child hit or caused the other to be upset? Based on proximity, your physical immediacy to another person, object, or location could lead to false perceptions about you and others. Perhaps these perceptions are harmless, but they can cause significant harm to one or more parties involved, particularly when it comes to how we manage our relationships with others. Asking questions and consistently examining our biases as we move through the world can help us avoid making assumptions.
Primacy and Recency Effects
It has frequently been said that you never get a second chance to make a first impression. Social psychologists use the term primacy effect to describe the phenomenon that people weigh what they see/learn first more heavily, seeing it as more salient than what comes later. To understand this further, let’s put the primacy effect into context with studies by Solomon Asch (1946) and Edward Jones (1968).
Asch (1946) conducted a study that asked participants to share their overall impression of a person based on a list of characteristics that he read off. All participants were given the same set of characteristics: intelligent, critical, impulsive, industrious, stubborn, envious. However, the order in which the characteristics were listed was changed. Asch's findings led him to conclude that when the list of characteristics started with the more positive traits, the participants were more likely to rate a person more positively. Conversely, when the negative characteristics were shared first, the overall impression was negative. In a related study by Jones (1968), participants were asked to watch one of two videotapes of a woman taking an intelligence test. Reading these studies, you might think that first is always best. The answer is much more complicated than that.
To demonstrate just how complex the human mind is, in some cases, the information that comes last is influential. Recency effect , though less common than primacy effect, explains the phenomenon that people give more weight to the last thing they see or hear. In a study by Wändi Bruine de Bruin (2005), higher marks were given to competitors who performed last in particular competitions (ice skating was one). So the question remains, what is best? As you might have guessed based on the theme of this chapter: it depends.
Now that we have a foundation for how influential similarities and differences, proximity, and primacy and recency can be in how we organize stimuli, let’s take a deeper dive into how we might interpret our perceptions.


8.1: Introduction
A human, like every sexually reproducing organism, begins life as a fertilized egg (embryo) or zygote. In our species, billions of cell divisions subsequently must occur in a controlled manner in order to produce a complex, multicellular human comprising trillions of cells. Thus, the original single-celled zygote is literally the ancestor of all cells in the body. However, once a human is fully grown, cell reproduction is still necessary to repair and regenerate tissues, and sometimes to increase our size! In fact, all multicellular organisms use cell division for growth and the maintenance and repair of cells and tissues. Cell division is closely regulated, and the occasional failure of this regulation can have life-threatening consequences. Single-celled organisms may also use cell division as their method of reproduction.
Contributors and Attributions
Remixed and/or curated from the following works:
Clark, M. A., Douglas, M., & Choi, J. (2018). 2.7.1 Introduction. In Biology 2e . OpenStax (CC BY 4.0; Access for free at https://openstax.org/books/biology-2e/pages/1-introduction ).


4.9: Sleep
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A newborn typically sleeps approximately 16.5 hours per 24-hour period. This is usually polyphasic sleep in that the infant is accumulating the 16.5 hours over several sleep periods throughout the day (Salkind, 2005). The infant is averaging 15 hours per 24-hour period by one month, and 14 hours by 6 months. By the time children turn two, they are averaging closer to 10 hours per 24 hours. Additionally, the average newborn will spend close to 50% of the sleep time in the Rapid Eye Movement (REM) phase, which decreases to 25% to 30% in childhood.70
Sudden Infant Death Syndrome and Safe Sleep
Sudden Infant Death Syndrome (SIDS) is identified when the death of a healthy infant occurs suddenly and unexpectedly, and medical and forensic investigation findings (including an autopsy) are inconclusive. SIDS is the leading cause of death in infants 1 to 12 months old, and approximately 1,500 infants died of SIDS in 2013 (CDC, 2015). Because SIDS is diagnosed when no other cause of death can be determined, possible causes of SIDS are regularly researched. One leading hypothesis suggests that infants who die from SIDS have abnormalities in the area of the brainstem responsible for regulating breathing (Weekes-Shackelford & Shackelford, 2005).71
Risk Factors Babies are at higher risk for SIDS if they:
- Sleep on their stomachs
- Sleep on soft surfaces, such as an adult mattress, couch, or chair or under soft coverings
- Sleep on or under soft or loose bedding
- Get too hot during sleep.
- Are exposed to cigarette smoke in the womb or in their environment, such as at home, in the car, in the bedroom, or other areas
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Sleep in an adult bed with parents, other children, or pets; this situation is especially dangerous if:
- The adult smokes, has recently had alcohol, or is tired.
- The baby is covered by a blanket or quilt.
- The baby sleeps with more than one bed-sharer.
- The baby is younger than 11 to 14 weeks of age.
Reducing the Risks There have been dramatic improvements in reducing baby deaths during sleep since the 1990s, when recommendations were introduced to place babies on their back for sleep. However, since the late 1990s, declines have slowed.
In 2012, the Back to Sleep campaign became the Safe to Sleep campaign. Safe to Sleep aims to educate all caregivers about SIDS and safe sleep practices. Current recommendations to reduce the risk of SIDS and other sleep-related causes of infant death:
- Always place baby on his or her back to sleep (for naps and at night).
- Use a firm and flat surface.
- Use only a tight fitting sheet on the sleep surface; no other bedding or soft items in the sleep area.
- Breastfeed.
- Share your room with a baby, but on a separate surface designed for infants (not your bed).
- Do not put soft objects, toys, crib bumpers, or loose bedding under, over, or anywhere near baby’s sleep area.
- Do no smoke during pregnancy or allow smoking around baby.
- Consider giving baby a pacifier.
- Do not let baby get too hot during sleep.
- Get regular health care (including vaccines).
- Avoid products that go against safe sleep recommendations, especially those that claim to prevent or reduce the risk of SIDS.
- Do not use heart or breathing monitors to reduce the risk of SIDS. 74
Contributors and Attributions
70. Lifespan Development: A Psychological Perspective by Martha Lally and Suzanne Valentine-French is licensed under CC BY-NC-SA 3.0
71. Lifespan Development: A Psychological Perspective by Martha Lally and Suzanne Valentine-French is licensed under CC BY-NC-SA 3.0
74. Ways to Reduce the Risk of SIDS and Other Sleep-Related Causes of Infant Death by the CDC is in the public domain


10: Your Social Problem
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- 10.2: Conduct research on your problem
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- 10.2.1: The Scientific Method in Practice
- 10.2.2: Data Collection and Analysis
- 10.2.3: Sociological Application - Using Research to Assess and Solve Problems
- 10.2.4: Reading Reports of Sociological Research
- 10.2.5: Being a Responsible Consumer of Research
- 10.2.6: Media Reports of Sociological Research
- 10.2.7: It’s Everywhere
- 10.2.8: Doing Research for a Cause
- 10.2.9: Appendix - Research Portfolio
- <IP_ADDRESS>: Instructions for completing the Research Portfolio assignment
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- 10.4: Develop an intervention
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- 10.4.1: Basic, Public, and Applied Sociology
- 10.4.2: Public Sociology
- 10.4.3: Levels of Analysis
- 10.4.4: Theoretical Approaches
- 10.4.5: Using a Sociological Perspective
- 10.4.6: References, Key Terms and Concepts
- 10.4.7: Skills and Competencies in Sociological Practice
- <IP_ADDRESS>: Practice Competencies
- 10.4.8: Community Interventions
- 10.4.9: Tools for Action
- <IP_ADDRESS>: Public Policy
- <IP_ADDRESS>: Community Organizing, Partnerships, and Coalitions
- <IP_ADDRESS>: Behavioral Community Approaches


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9.10: Chapter Review - Summary
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Summary
9.1
Microsoft Excel Basics
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A spreadsheet is a type of software that allows you to calculate mathematical equations and design tables that you can use for financial documents and that performs data analysis and mathematical calculations.
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The GUI of Excel is designed for manipulating the cells of the spreadsheet. The GUI also has a ribbon, which is a wide, multitabbed menu that contains features for customizing and formatting tables.
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The spreadsheet file is called a workbook and is composed of worksheets. A workbook can contain only one worksheet. However, you cannot have a worksheet without a workbook.
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Add new worksheets to a workbook by clicking on the + next to the sheet tabs found along the bottom of the interface.
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The Print feature of Excel allows you to print a single worksheet or the whole workbook.
9.2
Text and Numbers in Microsoft Excel
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Excel is a powerful tool to manage small to somewhat large sets of data.
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Data can be collected in Excel and analyzed to track a business’s sales or costs or a machine’s output, for example.
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Both text and numbers can be formatted to match the contents of the data such as formatting a description as text or a cost as accounting.
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The Page Layout tab includes commands to help you format the printed page in a professional manner.
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The Review tab is especially useful when sharing the spreadsheet for collaboration and ensuring accuracy in spelling and grammar.
9.3
Calculations and Basic Formulas in Microsoft Excel
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Excel uses the formula syntax to add, subtract, multiply, or divide. The formula syntax is comparable to writing out math equations by hand, with numbers and mathematical operators.
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The function syntax can perform the same calculations as the formula syntax for addition and multiplication but not for subtraction or division. The function syntax uses function names with cell references or ranges in parentheses to calculate.
9.4
Formatting and Templates in Microsoft Excel
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Formatting cells using colors, fonts, and borders can make a worksheet more readable and visually appealing. Other commands such as Merge Cells and Wrap Text can be used to adjust the information in the cells to enhance the professional presentation of the spreadsheet.
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By hiding columns or rows, you can make a more manageable worksheet or control what other users see and how they interact with it.
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Freezing rows or columns keeps the information visible when navigating large spreadsheets.
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Using conditional formatting identifies trends in the data or highlights values that are inside or outside a certain range.
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There are a variety of templates available within Excel and online for many uses such as invoicing, sales reports, and budgeting.
9.5
Google Sheets Basics
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Sheets serves many of the same purposes as Excel, but some of its tools are limited, compared with Excel. Many of the features are the same as or similar to Excel.
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The toolbar on Sheets is called the action bar. It is just one row of commands, unlike the wider Microsoft Office ribbon, with menus that are essentially the same as the tools in Excel, but the action bar also includes a row of buttons for frequently used commands.
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The process of creating a worksheet and formatting cells in an existing worksheet in Sheets is similar to that of Excel, with drop-down menus for common commands.
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Printing in Sheets is similar to printing in Excel.
9.6
Text and Numbers in Google Sheets
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The copy-and-paste function in Sheets works the same way as in Excel and allows the user to use the structure of an existing table to create a new one. Users can then manipulate the table by inserting columns or rows.
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To create a new table, first create header rows and then format the cells below it based on their contents. Then, as you add text or numbers, Sheets will automatically format them based on the type of data.
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Basic graphs require selecting the data for analysis and then inserting a graph. Sheets chooses the type of graph based on the type of data. You can change the graph type using the sidebar.
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Sheets has many of the same page layout and review features as Excel, with the major difference being the location of the tools in the interface.
9.7
Calculations and Basic Formulas in Google Sheets
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In Sheets, mathematical calculations such as addition, subtraction, multiplication, and division can use formulas, mathematical operators, and/or functions.
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The functions in Sheets differ some from Excel. Sheets has a few functions such as MINUS and DIVIDE that are not in Excel.
9.8
Formatting and Templates in Google Sheets
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Sheets has many of the same formatting functionalities as Excel, such as colors, fonts, borders, text wrapping, and merging cells.
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The formatting features allow for designing professional column and row headers. Hiding and unhiding columns or rows can simplify a worksheet to make it easier to use.
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Conditional formatting works the same as in Excel.
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Sheets contains some templates on the Welcome screen, but users of Sheets also share their own. Searching among only those publicly available templates can help you find many additional templates.


1.6: Racial Violence in Reconstruction
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Violence shattered the dream of biracial democracy. Still steeped in the violence of slavery, white southerners could scarcely imagine black free labor. Congressional investigator Carl Schurz reported that in the summer of 1865, southerners shared a near unanimous sentiment that “You cannot make the negro work, without physical compulsion.” 30 Violence had been used in the antebellum period to enforce slave labor and to define racial difference. In the post-emancipation period it was used to stifle black advancement and return to the old order.
Much of life in the antebellum South had been premised on slavery. The social order rested on a subjugated underclass, and the labor system required unfree laborers.  Whites were understood as fit for freedom and citizenship, blacks for chattel slave labor. The Confederate surrender at Appomattox Court House and the subsequent adoption by the U.S. Congress of the Thirteenth Amendment destroyed the institution of American slavery and threw southern society into disarray. The foundation of southern society had been shaken, but southern whites used black codes and racial terrorism to reassert control of former slaves.
Racial violence in the Reconstruction period took three major forms: riots against black political authority, interpersonal fights, and organized vigilante groups. There were riots in southern cities several times during Reconstruction. The most notable were the riots in Memphis and New Orleans in 1866, but other large-scale urban conflicts erupted in places including Laurens, South Carolina, in 1870; Colfax, Louisiana, in 1873; another in New Orleans in 1874; Yazoo City, Mississippi, in 1875; and Hamburg, South Carolina, in 1876. Southern cities grew rapidly after the war as migrants from the countryside—particularly freed slaves—flocked to urban centers. Cities became centers of Republican control. But white conservatives chafed at the influx of black residents and the establishment of biracial politics. In nearly every conflict, white conservatives initiated violence in reaction to Republican rallies or conventions or elections in which black men were to vote. The death tolls of these conflicts remain incalculable, and victims were overwhelmingly black.
Even everyday violence between individuals disproportionally targeted African Americans during Reconstruction. African Americans gained citizenship rights like the ability to serve on juries as a result of the Civil Rights Act of 1866 and the Fourteenth Amendment. But southern white men were almost never prosecuted for violence against black victims. White men beat or shot black men with relative impunity, and did so over minor squabbles, labor disputes, long-standing grudges, and crimes of passion. These incidents sometimes were reported to local federal authorities like the army or the Freedmen’s Bureau, but more often than not such violence was unreported and unprosecuted. 31
The violence committed by organized vigilante groups, sometimes called nightriders or bushwhackers, was more often premeditated. Groups of nightriders operated under cover of darkness and wore disguises to curtail black political involvement. Nightriders harassed and killed black candidates and officeholders and frightened voters away from the polls. They also aimed to limit black economic mobility by terrorizing freedpeople who tried to purchase land or otherwise become too independent from the white masters they used to rely on. They were terrorists and vigilantes, determined to stop the erosion of the antebellum South, and they were widespread and numerous, operating throughout the South. The Ku Klux Klan emerged in the late 1860s as the most infamous of these groups.
The Ku Klux Klan (KKK) was organized in 1866 in Pulaski, Tennessee, and had spread to nearly every state of the former Confederacy by 1868. The Klan drew heavily from the antebellum southern elite, but Klan groups sometimes overlapped with criminal gangs or former Confederate guerrilla groups. The Klan’s reputation became so potent, and its violence so widespread, that many groups not formally associated with it were called Ku Kluxers, and to “Ku Klux” meant to commit vigilante violence. While it is difficult to differentiate Klan actions from those of similar groups, such as the White Line, the Knights of the White Camellia, and the White Brotherhood, the distinctions hardly matter. All such groups were part of a web of terror that spread throughout the South during Reconstruction. In Panola County, Mississippi, between August 1870 and December 1872, twenty-four Klan-style murders occurred. And nearby, in Lafayette County, Klansmen drowned thirty black Mississippians in a single mass murder. Sometimes the violence was aimed at black men or women who had tried to buy land or dared to be insolent toward a white southerner. Other times, as with the beating of Republican sheriff and tax collector Allen Huggins, the Klan targeted white politicians who supported freedpeople’s civil rights. Numerous Republican politicians, perhaps dozens, were killed, either while in office or while campaigning. Thousands of individual citizens, men and women, white and black, had their homes raided and were whipped, raped, or murdered. 32
The federal government responded to southern paramilitary tactics by passing the Enforcement Acts between 1870 and 1871. The acts made it criminal to deprive African Americans of their civil rights. The acts also deemed violent Klan behavior as acts of rebellion against the United States and allowed for the use of U.S. troops to protect freedpeople. For a time, the federal government, its courts, and its troops, sought to put an end to the KKK and related groups. But the violence continued. By 1876, as southern Democrats reestablished “home rule” and “redeemed” the South from Republicans, federal opposition to the KKK weakened. National attention shifted away from the South and the activities of the Klan, but African Americans remained trapped in a world of white supremacy that restricted their economic, social, and political rights.
White conservatives would assert that Republicans, in denouncing violence, were “waving a bloody shirt” for political opportunity. The violence, according to many white conservatives, was fabricated, or not as bad as it was claimed, or an unavoidable consequence of the enfranchisement of African Americans. On December 22, 1871, R. Latham of Yorkville, South Carolina, wrote to the New York Tribune , voicing the beliefs of many white southerners as he declared that “the same principle that prompted the white men at Boston, disguised as Indians, to board, during the darkness of night, a vessel with tea, and throw her cargo into the Bay, clothed some of our people in Ku Klux gowns, and sent them out on missions technically illegal. Did the Ku Klux do wrong? You are ready to say they did and we will not argue the point with you. . . . Under the peculiar circumstances what could the people of South Carolina do but resort to Ku Kluxing?” 33
Victims and witnesses to the violence told a different story. Sallie Adkins of Warren County, Georgia, was traveling with her husband, Joseph, a Georgia state senator, when he was assassinated by Klansmen on May 10, 1869. She wrote President Ulysses S. Grant, asking for both physical protection and justice. “I am no Statesman,” she disclaimed; “I am only a poor woman whose husband has been murdered for his devotion to his country. I may have very foolish ideas of Government, States & Constitutions. But I feel that I have claims upon my country. The Rebels imprisoned my Husband. Pardoned Rebels murdered him. There is no law for the punishment of them who do deeds of this sort. . . . I demand that you, President Grant, keep the pledge you made the nation—make it safe for any man to utter boldly and openly his devotion to the United States.” 34
The political and social consequences of the violence were as lasting as the physical and mental trauma suffered by victims and witnesses. Terrorism worked to end federal involvement in Reconstruction and helped to usher in a new era of racial repression.
African Americans actively sought ways to shed the vestiges of slavery. Many discarded the names their former masters had chosen for them and adopted new names like “Freeman” and “Lincoln” that affirmed their new identities as free citizens. Others resettled far from their former plantations, hoping to eventually farm their own land or run their own businesses. By the end of Reconstruction, the desire for self-definition, economic independence, and racial pride coalesced in the founding of dozens of black towns across the South. Perhaps the most well-known of these towns was Mound Bayou, Mississippi, a Delta town established in 1887 by Isaiah Montgomery and Ben Green, former slaves of Joseph and Jefferson Davis. Residents of the town took pride in the fact that African Americans owned all of the property in town, including banks, insurance companies, shops, and the surrounding farms. The town celebrated African American cultural and economic achievements during their annual festival, Mound Bayou Days. These tight-knit communities provided African Americans with spaces where they could live free from the indignities of segregation and the exploitation of sharecropping on white-owned plantations. 35


3.7: Light Scattering by a Slit
One of the most spectacular successes for phasor analysis arises in the study of light diffraction by a narrow slit. The experiment is to shine laser light through a slit in an otherwise opaque sheet and observe the pattern of light that falls on a distant screen. When the slit is very narrow compared with the wavelength of the light, then the light that falls on the screen is nearly uniform in intensity. However, when the width of the slit is comparable to the wavelength of the light, the pattern of light that falls on the screen is scalloped in intensity, showing alternating light and dark bands. The experiment, and the observed results, are illustrated in the Figure.
Why should this experiment produce this result? Phasor analysis illuminates the question and produces an elegant mathematical description of a profoundly important optical experiment.
Huygens's Principle
We will assume, as Christiaan Huygens did, that the light incident on the slit sets up a light field in the slit that may be modeled by \(N\) discrete sources, each of which radiates a “spherical wave of light.” This model is illustrated in the Figure. The distance between sources is \(d\), and \(Nd=L\) is the width of the slit. Each source is indexed by \(n\), and \(n\) runs from 0 to \(N−1\). The 0 th source is located at the origin of our coordinate system.
The spherical wave radiated by the \(n^{\mathrm{th}}\) source is described by the equation
\[E(r_n,t)=\mathrm{Re}\{\frac A N e^{j[ωt−(2π/λ)r_n]}\} \nonumber \]
The function \(E(r_n,t)\) describes the “electric field” at time \(t\) and distance \(r_n\) from the \(n^\mathrm{th}\) source. The field is constant as long as the variable \(ωt−(2π/λ)r_n\) is constant. Therefore, if we freeze time at \(t=t_0\), the field will be constant on a sphere of radius \(r_n\). This is illustrated in the Figure.
Fix \(r_n\) in \(E(r_n,t)\) and show that \(E(r_n,t)\) varies cosinusoidally with time \(t\). Sketch the function and interpret it. What is its period?
Fix \(t\) in \(E(r_n,t)\) and show that \(E(r_n,t)\) varies cosinusoidally with radius \(r_n\). Sketch the function and interpret it. Call the “period in \(r_n\)" the “wavelength.” Show that the wavelength is \(λ\).
The “crest of the wave \(E(r_n,t)\)" occurs when \(ωt−(2π/λ)r_n=0\). Show that the crest moves through space at velocity \(v=ωλ/2π\).
Geometry
If we now pick a point \(P\) on a distant screen, that point will be at distance \(r_0\) from source 0,...,\(r_n\) from source \(n\),..., and so on. If we isolate the sources 0 and \(n\), then we have the geometric picture of the Figure. The angle \(φ\) is the angle that point \(P\) makes with the horizontal axis. The Pythagorean theorem says that the connection between distances \(r_0\) and \(r_n\) is
\[(r_n−nd\sinφ)^2+(nd\cosφ)^2=r^2_0 \nonumber \]
Let's try the solution
\[r_n=r_0+nd\sinφ \nonumber \]
This solution produces the approximate identity
\[r^2_0+(nd\cosφ)^2≅r^2_0 \nonumber \]
\[r^2_0+(nd\cosφ)^2≅r^2_0 \nonumber \]
\[1+(\frac {nd} {r_0} \cosφ)^2≅1 \nonumber \]
This will be close for \(\frac {nd} {r_0} <<1\). We will assume that the slit width \(L\) is small compared to the distance to any point on the screen.
Then \(\frac {Nd} {r_0} = \frac {L} {r_0} << 1\), in which case the approximate solution for \(r_n\) is valid for all \(n\). This means that, for any point \(P\) on the distant screen, the light contributed by the \(n^{th}\) source is approximately
\[E_n(φ,t)=\mathrm{Re}\{\frac A N e^{j[ωt−(2π/λ)(r_0+nd\sinφ)]}\}=\mathrm{Re}\{\frac A N e^{−j(2π/λ)r_0}e^{−j(2π/λ)nd\sinφ}e^{jωt}\} \nonumber \]
The phasor representation for this function is just
\[E_n(φ)=\frac A N e^{−j(2π/λ)r_0} e^{−j(2π/λ)nd\sinφ} \nonumber \]
Note that \(E_0(φ)\), the phasor associated with the 0 th source, is \(\frac A N e^{−j(2π/λ)r_0}\). Therefore we may write the phasor representation for the light contributed by the \(n^{th}\) source to be
\[E_n(φ)=E_0(φ)e^{−j(2π/λ)nd\sinφ} \nonumber \]
This result is very important because it shows the light arriving at point \(P\) from different sources to be “out of phase” by an amount that depends on the ratio \(\frac {nd\sinφ} λ\)
Phasors and Interference
The phasor representation for the field observed at point \(P\) on the screen is the sum of the phasors contributed by each source:
\[E(φ)=∑_{n=0}^{N−1} E_n(φ)=E_0(φ)∑_{n=0}^{N−1}e^{−j(2π/λ)nd\sinφ} \nonumber \]
This is a sum of the form
\[E(φ)=E_0(φ)∑_{n=0}^{N−1}e^{jnθ} \nonumber \]
where the angle \(θ\) is \((2π/λ)d\sinφ\). This sum is illustrated in the Figure for several representative values of \(θ\). Note that for small \(θ\), meaning small \(φ\), the sum has large magnitude, whereas for \(θ\) on the order of \(2π/N\), the sum is small. This simple geometric interpretation shows that for some values of \(φ\), corresponding to some points \(P\) on the screen, there will be constructive interference between the phasors, while for other values of \(φ\) there will be destructive interference. Constructive interference produces bright light, and destructive interference produces darkness.
The geometry of the Figure is illuminating. However, we already know from our study of complex numbers and geometric sums that the phasor sum of the Equation may be written as
\[E(φ)=E_0(φ)\frac {1−e^{−j(2π/λ)Nd\sinφ}} {1−e^{−j(2π/λ)d\sinφ}} \nonumber \]
This result may be manipulated to produce the form
\[E(φ)=\frac A N e^{−j(2π/λ)r_0} e^{−j(π/λ)(N−1)d\sinφ}\frac {\sin(Nπ\frac d λ \sinφ)} {\sin(π\frac d λ \sinφ)} \nonumber \]
The magnitude is the intensity of the light at angle \(φ\) from horizontal:
\[|E(φ)|=|\frac A N \frac {\sin(Nπ\frac d λ \sinφ)} {\sin(π\frac d λ \sinφ)}| \nonumber \]
Derive \(\frac A N e^{−j(2π/λ)r_0} e^{−j(π/λ)(N−1)d\sinφ}\frac {\sin(Nπ\frac d λ \sinφ)} {\sin(π\frac d λ \sinφ)}\) from \(E_0(φ)\frac {1−e^{−j(2π/λ)Nd\sinφ}} {1−e^{−j(2π/λ)d\sinφ}}\).
Limiting Form
Huygens's model is exact when d shrinks to 0 and N increases to infinity in such a way that Nd→L, the slit width. Then
\[|E(φ)|→|\frac {A\sin(\frac {πL} λ \sinφ)} {\frac {πL} λ \sinφ}| \nonumber \]
This function is plotted in theFigure for two values of \(\frac L λ\) the width of the slit measured in wavelengths. When \(\frac L λ<<1\) (i.e., \(/frac λ L >>1\)), then the light is uniformly distributed on the screen. However, when \(\frac L λ>1(\frac λ L<1)\), then the function has many zeros for \(\frac| \sin φ|<1\), as illustrated in the figure. These zeros correspond to dark spots on the screen where the fields radiated from the infinity of points within the slit interfere destructively.
 Compare with the continuous, limiting form
\(∣\frac {A\sin(π\frac L λ \sinφ)} {π\frac L λ \sinφ}∣\)


2.2: Events and Random Variables
The purpose of this section is to study two basic types of objects that form part of the model of a random experiment. If you are a new student of probability, just ignore the measure-theoretic terminology and skip the technical details.
Sample Spaces
The Set of Outcomes
Recall that in a random experiment, the outcome cannot be predicted with certainty, before the experiment is run. On the other hand:
We assume that we can identify a fixed set \( S \) that includes all possible outcomes of a random experiment. This set plays the role of the universal set when modeling the experiment.
For simple experiments, \( S \) may be precisely the set of possible outcomes. More often, for complex experiments, \( S \) is a mathematically convenient set that includes the possible outcomes and perhaps other elements as well. For example, if the experiment is to throw a standard die and record the score that occurs, we would let \(S = \{1, 2, 3, 4, 5, 6\}\), the set of possible outcomes. On the other hand, if the experiment is to capture a cicada and measure its body weight (in milligrams), we might conveniently take \(S = [0, \infty)\), even though most elements of this set are impossible (we hope!). The problem is that we may not know exactly the outcomes that are possible. Can a light bulb burn without failure for one thousand hours? For one thousand days? for one thousand years?
Often the outcome of a random experiment consists of one or more real measurements, and thus, the \( S \) consists of all possible measurement sequences , a subset of \(\R^n\) for some \(n \in \N_+\). More generally, suppose that we have \(n\) experiments and that \( S_i \) is the set of outcomes for experiment \( i \in \{1, 2, \ldots, n\} \). Then the Cartesian product \(S_1 \times S_2 \times \cdots \times S_n\) is the natural set of outcomes for the compound experiment that consists of performing the \(n\) experiments in sequence. In particular, if we have a basic experiment with \(S\) as the set of outcomes, then \(S^n\) is the natural set of outcomes for the compound experiment that consists of \(n\) replications of the basic experiment. Similarly, if we have an infinite sequence of experiments and \( S_i \) is the set of outcomes for experiment \( i \in \N_+ \), then then \(S_1 \times S_2 \times \cdots\) is the natural set of outcomes for the compound experiment that consists of performing the given experiments in sequence. In particular, the set of outcomes for the compound experiment that consists of indefinite replications of a basic experiment is \(S^\infty = S \times S \times \cdots \). This is an essential special case, because (classical) probability theory is based on the idea of replicating a given experiment.
Events
Consider again a random experiment with \( S \) as the set of outcomes. Certain subsets of \( S \) are referred to as events . Suppose that \( A \subseteq S \) is a given event, and that the experiment is run, resulting in outcome \( s \in S \).
- If \( s \in A \) then we say that \( A \) occurs .
- If \( s \notin A \) then we say that \( A \) does not occur .
Intuitively, you should think of an event as a meaningful statement about the experiment: every such statement translates into an event, namely the set of outcomes for which the statement is true. In particular, \(S\) itself is an event; by definition it always occurs. At the other extreme, the empty set \(\emptyset\) is also an event; by definition it never occurs.
For a note on terminology, recall that a mathematical space consists of a set together with other mathematical structures defined on the set. An example you may be familiar with is a vector space , which consists of a set (the vectors) together with the operations of addition and scalar multiplication. In probability theory, many authors use the term sample space for the set of outcomes of a random experiment, but here is the more careful definition:
The sample space of an experiment is \( (S, \mathscr S) \) where \( S \) is the set of outcomes and \( \mathscr S \) is the collection of events.
Details
Sometimes not every subset of \(S\) can be allowed as an event, but the collection of events \( \mathscr S \) is required to be a \( \sigma \)-algebra , so that the sample space \( (S, \mathscr S) \) is a measurable space. The axioms of a \(\sigma\)-algebra ensure that new sets that are constructed in a reasonable way from given events, using the set operations, are themselves valid events. Most of the sample spaces that occur in elementary probability fall into two general categories.
- Discrete : \( S \) is countable and \( \mathscr S = \mathscr P(S) \) is the collection of all subsets of \( S \). In this case, the sample space \((S, \mathscr S)\) is discrete .
- Euclidean : \(S\) is a measurable subset of \(\R^n\) for some \(n \in \N_+\) and \(\mathscr S\) is the collection of measurable subsets of \(S\).
In (b), the mearuable subsets of \(\R^n\) include all of the sets encountered in calculus and in standard applications of probability theory, and many more besides. Nonetheless, for technical reasons, certain very weird subsets must be excluded. Typically \( S \) is a set defined by a finite number of inequalities involving elementary functions.
The Algebra of Events
The standard algebra of sets leads to a grammar for discussing random experiments and allows us to construct new events from given events. In the following results, suppose that \( S \) is the set of outcomes of a random experiment, and that \(A\) and \(B\) are events.
\(A \subseteq B\) if and only if the occurrence of \(A\) implies the occurrence of \(B\).
Proof
Recall that \( \subseteq \) is the subset relation. So by definition, \( A \subseteq B \) means that \( s \in A \) implies \( s \in B \).
\(A \cup B\) is the event that occurs if and only if \(A\) occurs or \(B\) occurs.
Proof
Recall that \( A \cup B \) is the union of \( A \) and \( B \). So by defintion, \( s \in A \cup B \) if and only if \( s \in A \) or \( s \in B \).
\(A \cap B\) is the event that occurs if and only if \(A\) occurs and \(B\) occurs.
Proof
Recall that \( A \cap B \) is the intersection of \( A \) and \( B \). So by definiton, \( s \in A \cap B \) if and only if \( s \in A \) and \( s \in B \).
\(A\) and \(B\) are disjoint if and only if they are mutually exclusive ; they cannot both occur on the same run of the experiment.
Proof
By definition, \( A \) and \( B \) disjoint means that \( A \cap B = \emptyset \).
\(A^c\) is the event that occurs if and only if \(A\) does not occur.
Proof
Recall that \( A^c \) is the complement of \( A \), so \( s \in A^c \) if and only if \( s \notin A \).
\(A \setminus B\) is the event that occurs if and only if \(A\) occurs and \(B\) does not occur.
Proof
Recall that \( A \setminus B = A \cap B^c \). Hence \( s \in A \setminus B \) if and only if \( s \in A \) and \( s \notin B \).
\((A \cap B^c) \cup (B \cap A^c)\) is the event that occurs if and only if one but not both of the given events occurs.
Proof
The events in the union are disjoint. So for \( s \) is in the given event if and only if either \( s \in A \) and \( s \notin B \), or \( s \in B \) and \( s \notin A \).
Recall that the event in (10) is the symmetric difference of \(A\) and \(B\), and is sometimes denoted \(A \Delta B\). This event corresponds to exclusive or , as opposed to the ordinary union \( A \cup B \) which corresponds to inclusive or .
\((A \cap B) \cup (A^c \cap B^c)\) is the event that occurs if and only if both or neither of the given events occurs.
Proof
The events in the union are disjoint. Thus \( s \) is in the given event if and only if either \( s \in A \) and \( s \in B \), or \( s \notin A \) and \( s \notin B \).
In the Venn diagram app, observe the diagram of each of the 16 events that can be constructed from \(A\) and \(B\).
Suppose now that \(\mathscr{A} = \{A_i: i \in I\}\) is a collection of events for the random experiment, where \(I\) is a countable index set.
\( \bigcup \mathscr{A} = \bigcup_{i \in I} A_i \) is the event that occurs if and only if at least one event in the collection occurs.
Proof
Note that \( s \in \bigcup_{i \in I} A_i \) if and only if \( s \in A_i \) for some \( i \in I \).
\( \bigcap \mathscr{A} = \bigcap_{i \in I} A_i \) is the event that occurs if and only if every event in the collection occurs:
Proof
Note that \( s \in \bigcap_{i \in I} A_i \) if and only if \( s \in A_i \) for every \( i \in I \).
\(\mathscr{A}\) is a pairwise disjoint collection if and only if the events are mutually exclusive ; at most one of the events could occur on a given run of the experiment.
Proof
By definition, \( A_i \cap A_j = \emptyset \) for distinct \( i, \, j \in I \).
Suppose now that \((A_1, A_2, \ldots\)) is an infinite sequence of events.
\(\bigcap_{n=1}^\infty \bigcup_{i=n}^\infty A_i\) is the event that occurs if and only if infinitely many of the given events occur. This event is sometimes called the limit superior of \((A_1, A_2, \ldots)\).
Proof
Note that \( s \) is in the given event if and only if for every \( n \in \N_+ \) there exists \( i \in \N_+ \) with \( i \ge n \) such that \( s \in A_i \). In turn this means that \( s \in A_i \) for infinitely many \( i \in I \).
\(\bigcup_{n=1}^\infty \bigcap_{i=n}^\infty A_i\) is the event that occurs if and only if all but finitely many of the given events occur. This event is sometimes called the limit inferior of \((A_1, A_2, \ldots)\).
Proof
Note that \( s \) is in the given event if and only if there exists \( n \in \N_+ \) such that \( s \in A_i \) for every \( i \in \N_+ \) with \( i \ge n \). In turn, this means that \( s \in A_i \) for all but finitely many \( i \in I \).
Limit superiors and inferiors are discussed in more detail in the section on convergence.
Random Variables
Intuitively, a random variable is a measurement of interest in the context of the experiment. Simple examples include the number of heads when a coin is tossed several times, the sum of the scores when a pair of dice are thrown, the lifetime of a device subject to random stress, the weight of a person chosen from a population. Many more examples are given below in the exercises below . Mathematically, a random variable is a function defined on the set of outcomes.
A function \( X \) from \( S \) into a set \( T \) is a random variable for the experiment with values in \( T \).
Details
The set \( T \) will also come with a \( \sigma \)-algebra \( \mathscr T \) of admissible subsets, so that \((T, \mathscr T)\) is a measurable space, just like \( (S, \mathscr S) \). The function \( X \) is required to be measurable, an assumption which ensures that meaningful statements involving \( X \) define events. In the discussion below, all subsets of \( T \) are assumed to be in \( \mathscr T \).
Probability has its own notation, very different from other branches of mathematics. As a case in point, random variables, even though they are functions, are usually denoted by capital letters near the end of the alphabet. The use of a letter near the end of the alphabet is intended to emphasize the idea that the object is a variable in the context of the experiment. The use of a capital letter is intended to emphasize the fact that it is not an ordinary algebraic variable to which we can assign a specific value, but rather a random variable whose value is indeterminate until we run the experiment. Specifically, when we run the experiment an outcome \(s \in S\) occurs, and random variable \(X\) takes the value \(X(s) \in T\).
If \( B \subseteq T \), we use the notation \( \{X \in B\} \) for the inverse image \( \{s \in S: X(s) \in B\} \), rather than \( X^{-1}(B) \). Again, the notation is more natural since we think of \( X \) as a variable in the experiment. Think of \( \{X \in B\} \) as a statement about \( X \), which then translates into the event \( \{s \in S: X(s) \in B\} \)
Again, every statement about a random variable \( X \) with values in \( T \) translates into an inverse image of the form \( \{X \in B\} \) for some \( B \in \mathscr T \). So, for example, if \( x \in T \) then \(\{X = x\} = \{X \in \{x\}\} = \left\{s \in S: X(s) = x\right\}\). If \(X\) is a real-valued random variable and \( a, \, b \in \R \) with \( a \lt b \) then \(\{ a \leq X \leq b\} = \left\{ X \in [a, b]\right\} = \{s \in S: a \leq X(s) \leq b\}\).
Suppose that \(X\) is a random variable taking values in \(T\), and that \(A, \, B \subseteq T\). Then
- \(\{X \in A \cup B\} = \{X \in A\} \cup \{X \in B\}\)
- \(\{X \in A \cap B\} = \{X \in A\} \cap \{X \in B\}\)
- \(\{X \in A \setminus B\} = \{X \in A\} \setminus \{X \in B\}\)
- \(A \subseteq B \implies \{X \in A\} \subseteq \{X \in B\}\)
- If \(A\) and \(B\) are disjoint, then so are \(\{X \in A\}\) and \(\{X \in B\}\).
Proof
This is a restatement of the fact that inverse images of a function preserve the set operations; only the notation changes (and is simpler).
- \(s \in \{X \in A \cup B\}\) if and only if \(X(s) \in A \cup B\) if and only if \(X(s) \in A\) or \(X(s) \in B\) if and only if \(s \in \{X \in A\}\) or \(s \in \{X \in B\}\) if and only if \(s \in \{X \in A\} \cup \{X \in B\}\).
- The proof is exactly the same as (a), with and replacing or .
- The proof is also exactly the same as (a), with but not replacing or .
- If \(s \in \{X \in A\}\) then \(X(s) \in A\) so \(X(s) \in B\) and hence \(s \in \{X \in B\}\).
- This follows from part (b).
As with a general function, the result in part (a) holds for the union of a countable collection of subsets, and the result in part (b) holds for the intersection of a countable collection of subsets. No new ideas are involved; only the notation is more complicated.
Often, a random variable takes values in a subset \(T\) of \(\R^k\) for some \(k \in \N_+\). We might express such a random variable as \(\bs{X} = (X_1, X_2, \ldots, X_k)\) where \(X_i\) is a real-valued random variable for each \(i \in \{1, 2, \ldots, k\}\). In this case, we usually refer to \(\bs{X}\) as a random vector , to emphasize its higher-dimensional character. A random variable can have an even more complicated structure. For example, if the experiment is to select \(n\) objects from a population and record various real measurements for each object, then the outcome of the experiment is a vector of vectors: \(\bs{X} = (X_1, X_2, \ldots, X_n)\) where \(X_i\) is the vector of measurements for the \(i\)th object. There are other possibilities; a random variable could be an infinite sequence, or could be set-valued. Specific examples are given in the computational exercises below. However, the important point is simply that a random variable is a function defined on the set of outcomes \(S\).
The outcome of the experiment itself can be thought of as a random variable. Specifically, let \(T = S\) and let \(X\) denote the identify function on \(S\) so that \(X(s) = s\) for \(s \in S\). Then trivially \(X\) is a random variable, and the events that can be defined in terms of \(X\) are simply the original events of the experiment. That is, if \( A \) is an event then \( \{X \in A\} = A \). Conversely, every random variable effectively defines a new random experiment.
In the general setting above, a random variable \(X\) defines a new random experiment with \( T \) as the new set of outcomes and subsets of \( T \) as the new collection of events.
Details
Technically, the \( \sigma \)-algebra \( \mathscr T \) would be the new collection of events.
In fact, often a random experiment is modeled by specifying the random variables of interest, in the language of the experiment. Then, a mathematical definition of the random variables specifies the sample space. A function (or transformation) of a random variable defines a new random variable.
Suppose that \( X \) is a random variable for the experiment with values in \(T\) and that \( g \) is a function from \( T \) into another set \( U \). Then \( Y = g(X) \) is a random variable with values in \( U \).
Details
Technically, \( T \) and \( U \) both come with \(\sigma\)-algebras of admissible subsets \( \mathscr T \) and \( \mathscr U \), respectively. The function \( g \), just like the function \( X \), is required to be measurable. This assumption ensures that \(Y = g(X)\) is a measurable function from \(S\) into \(U\), and hence is a valid random variable.
Note that, as functions, \( g(X) = g \circ X \), the composition of \( g \) with \( X \). But again, thinking of \( X \) and \( Y \) as variables in the context of the experiment, the notation \( Y = g(X) \) is much more natural.
Indicator Variables
For an event \(A\), the indicator function of \(A\) is called the indicator variable of \(A\).
The value of this random variables tells us whether or not \(A\) has occurred: \[ \bs{1}_A = \begin{cases} 1, & A \text{ occurs} \\ 0, & A \text{ does not occur} \end{cases} \] That is, as a function on \( S \), \[ \bs{1}_A(s) = \begin{cases} 1, & s \in A \\ 0, & s \notin A \end{cases} \]
If \(X\) is a random variable that takes values 0 and 1, then \(X\) is the indicator variable of the event \(\{X = 1\}\).
Proof
Note that for \( s \in S \), \( X(s) = 1\) if \( s \in \{X = 1\} \) and \( X(s) = 0 \) otherwise.
Recall also that the set algebra of events translates into the arithmetic algebra of indicator variables.
Suppose that \(A\) and \(B\) are events.
- \(\bs{1}_{A \cap B} = \bs{1}_A \bs{1}_B = \min\left\{\bs{1}_A, \bs{1}_B\right\}\)
- \(\bs{1}_{A \cup B} = 1 - \left(1 - \bs{1}_A\right)\left(1 - \bs{1}_B\right) = \max\left\{\bs{1}_A, \bs{1}_B\right\}\)
- \(\bs{1}_{B \setminus A} = \bs{1}_B \left(1 - \bs{1}_A\right)\)
- \(\bs{1}_{A^c} = 1 - \bs{1}_A\)
- \(A \subseteq B\) if and only if \(\bs{1}_A \leq \bs{1}_B\)
The results in part (a) extends to arbitrary intersections and the results in part (b) extends to arbitrary unions. If the event \( A \) has a complicated description, sometimes we use \( \bs 1 (A) \) for the indicator variable rather that \( \bs 1_A \).
Examples and Applications
Recall that probability theory is often illustrated using simple devices from games of chance: coins, dice, cards, spinners, urns with balls, and so forth. Examples based on such devices are pedagogically valuable because of their simplicity and conceptual clarity. On the other hand, remember that probability is not only about gambling and games of chance. Rather, try to see problems involving coins, dice, etc. as metaphors for more complex and realistic problems.
Coins and Dice
The basic coin experiment consists of tossing a coin \(n\) times and recording the sequence of scores \((X_1, X_2, \ldots, X_n)\) (where 1 denotes heads and 0 denotes tails). This experiment is a generic example of \(n\) Bernoulli trials, named for Jacob Bernoulli.
Consider the coin experiment with \(n = 4\), and Let \(Y\) denote the number of heads.
- Give the set of outcomes \(S\) in list form.
- Give the event \(\{Y = k\}\) in list form for each \(k \in \{0, 1, 2, 3, 4\}\).
Answer
To simplify the notation, we represent outcomes a bit strings rather than ordered sequences.
- \(S = \{1111, 1110, 1101, 1011, 0111, 1100, 1010, 1001, 0110, 0101, 0011, 1000, 0100, 0010, 0001, 0000\}\)
- \begin{align} \{Y = 0\} & = \{0000\} \\ \{Y = 1\} & = \{1000, 0100, 0010, 0001\} \\ \{Y = 2\} & = \{1100, 1010, 1001, 0110, 0101, 0011\} \\ \{Y = 3\} & = \{1110, 1101, 1011, 0111\} \\ \{Y = 4\} & = \{1111\} \end{align}
In the simulation of the coin experiment, set \(n = 4\). Run the experiment 100 times and count the number of times that the event \(\{Y = 2\}\) occurs.
Now consider the general coin experiment with the coin tossed \(n\) times, and let \(Y\) denote the number of heads.
- Give the set of outcomes \(S\) in Cartesian product form, and give the cardinality of \(S\).
- Express \(Y\) as a function on \(S\).
- Find \(\#\{Y = k\}\) (as a subset of \(S\)) for \(k \in \{0, 1, \ldots, n\}\)
Answer
- \(S = \{0, 1\}^n\) and \(\#(S) = 2^n\).
- \(Y(x_1, x_2, \ldots, x_n) = x_1 + x_2 + \cdots + x_n\). The set of possible values is \(\{0, 1, \ldots, n\}\)
- \(\#\{Y = k\} = \binom{n}{k}\)
The basic dice experiment consists of throwing \(n\) distinct \(k\)-sided dice (with faces numbered from 1 to \(k\)) and recording the sequence of scores \((X_1, X_2, \ldots, X_n)\). This experiment is a generic example of \(n\) multinomial trials. The special case \(k = 6\) corresponds to standard dice .
Consider the dice experiment with \(n = 2\) standard dice. Let \(S\) denote the set of outcomes, \(A\) the event that the first die score is 1, and \(B\) the event that the sum of the scores is 7. Give each of the following events in the form indicated:
- \(S\) in Cartesian product form
- \(A\) in list form
- \(B\) in list form
- \(A \cup B\) in list form
- \(A \cap B\) in list form
- \(A^c \cap B^c\) in predicate form
Answer
- \(S = \{1, 2, 3, 4, 5, 6\}^2\)
- \(A = \{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6)\}\)
- \(B = \{(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\}\)
- \(A \cup B = \{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\}\)
- \(A \cap B = \{(1,6)\}\)
- \(A^c \cap B^c = \{(x, y) \in S: x + y \ne 7 \text{ and } x \ne 1\}\)
In the simulation of the dice experiment, set \(n = 2\). Run the experiment 100 times and count the number of times each event in the previous exercise occurs.
Consider the dice experiment with \(n = 2\) standard dice, and let \( S \) denote the set of outcomes, \(Y\) the sum of the scores, \(U\) the minimum score, and \(V\) the maximum score.
- Express \(Y\) as a function on \( S \) and give the set of possible values in list form.
- Express \(U\) as a function on \( S \) and give the set of possible values in list form.
- Express \(V\) as a function on the \( S \) and give the set of possible values in list form.
- Give the set of possible values of \((U, V)\) in predicate from
Answer
Note that \( S = \{1, 2, 3, 4, 5, 6\}^2 \). The following functions are defined on \( S \).
- \(Y(x_1, x_2) = x_1 + x_2\). The set of values is \(\{2, 3, \ldots, 12\}\)
- \(U(x_1, x_2) = \min\{x_1, x_2\}\). The set of values is \(\{1, 2, \ldots, 6\}\)
- \(V(x_1, x_2) = \max\{x_1, x_2\}\). The set of values is \(\{1, 2, \ldots, 6\}\)
- \(\left\{(u, v) \in \{1, 2, 3, 4, 5, 6\}^2: u \le v\right\}\)
Consider again the dice experiment with \(n = 2\) standard dice, and let \( S \) denote the set of outcomes, \(Y\) the sum of the scores, \(U\) the minimum score, and \(V\) the maximum score. Give each of the following as subsets of \( S \), in list form.
- \(\{X_1 \lt 3, X_2 \gt 4\}\)
- \(\{Y = 7\}\)
- \(\{U = 2\}\)
- \(\{V = 4\}\)
- \(\{U = V\}\)
Answer
- \(\{(1,5), (2,5), (1,6), (2,6)\}\)
- \(\{(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\}\)
- \(\{(2,2), (2,3), (3,2), (2,4), (4,2), (2,5), (5,2), (2,6), (6,2)\}\)
- \(\{(4,1), (1,4), (2,4), (4,2), (4,3), (3,4), (4,4)\}\)
- \(\{(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)\}\)
In the dice experiment, set \(n = 2\). Run the experiment 100 times. Count the number of times each event in the previous exercise occurred.
In the general dice experiment with \(n\) distinct \(k\)-sided dice, let \(Y\) denote the sum of the scores, \(U\) the minimum score, and \(V\) the maximum score.
- Give the set of outcomes \( S \) and find \( \#(S) \).
- Express \(Y\) as a function on \( S \), and give the set of possible values in list form.
- Express \(U\) as a function on \( S \), and give the set of possible values in list form.
- Express \(V\) as a function on \( S \), and give the set of possible values in list form.
- Give the set of possible values of \((U, V)\) in predicate from.
Answer
- \(S = \{1, 2, \ldots, k\}^n\) and \(\#(S) = k^n\)
- \(Y(x_1, x_2, \ldots, x_n) = x_1 + x_2 + \cdots + x_n\). The set of possible values is \(\{n, n + 1, \ldots, n k\}\)
- \(U(x_1, x_2, \ldots, x_n) = \min\{x_1, x_2, \ldots, x_n\}\). The set of possible values is \(\{1, 2, \ldots, k\}\).
- \(V(x_1, x_2, \ldots, x_n) = \max\{x_1, x_2 \ldots, x_n\}\). The set of possible values is \(\{1, 2, \ldots, k\}\)
- \(\left\{(u, v) \in \{1, 2, \ldots, k\}^2: u \le v\right\}\)
The set of outcomes of a random experiment depends of course on what information is recorded. The following exercise is an illustration.
An experiment consists of throwing a pair of standard dice repeatedly until the sum of the two scores is either 5 or 7. Let \(A\) denote the event that the sum is 5 rather than 7 on the final throw. Experiments of this type arise in the casino game craps.
- Suppose that the pair of scores on each throw is recorded. Define the set of outcomes of the experiment and describe \(A\) as a subset of this set.
- Suppose that the pair of scores on the final throw is recorded. Define the set of outcomes of the experiment and describe \(A\) as a subset of this set.
Answer
Let \(D_5 = \{(1,4), (2,3), (3,2), (4,1)\}\), \(D_7 = \{(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\}\), \(D = D_5 \cup D_7 \), and \( C = D^c \)
- \(S = D \cup (C \times D) \cup (C^2 \times D) \cup \cdots\), \(A = D_5 \cup (C \times D_5) \cup (C^2 \times D_5) \cup \cdots\)
- \(S = D\), \(A = D_5\)
Suppose that 3 standard dice are rolled and the sequence of scores \((X_1, X_2, X_3)\) is recorded. A person pays $1 to play. If some of the dice come up 6, then the player receives her $1 back, plus $1 for each 6. Otherwise she loses her $1. Let \(W\) denote the person's net winnings. This is the game of chuck-a-luck and is treated in more detail in the chapter on Games of Chance.
- Give the set of outcomes \(S\) in Cartesian product form.
- Express \(W\) as a function on \(S\) and give the set of possible values in list form.
Answer
- \(S = \{1, 2, 3, 4, 5, 6\}^3\)
- \(W(x_1, x_2, x_3) = \bs{1}\left(x_1 = 6\right) + \bs{1}\left(x_2 = 6\right) + \bs{1}\left(x_3 = 6\right) - \bs{1}\left(x_1 \ne 6, x_2 \ne 6, x_3 \ne 6\right)\). The set of possible values is \(\{-1, 1, 2, 3\}\)
Play the chuck-a-luck experiment a few times and see how you do.
In the die-coin experiment , a standard die is rolled and then a coin is tossed the number of times shown on the die. The sequence of coin scores \(\bs{X}\) is recorded (0 for tails, 1 for heads). Let \(N\) denote the die score and \(Y\) the number of heads.
- Give the set of outcomes \(S\) in terms of Cartesian powers and find \(\#(S)\).
- Express \(N\) as a function on \(S\) and give the set of possible values in list form.
- Express \(Y\) as a function on \(S\) and give the set of possible values in list form.
- Give the event \(A\) that all tosses result in heads in list form.
Answer
- \(S = \bigcup_{n=1}^6 \{0, 1\}^n\), \(\#(S) = 126\)
- \(N(x_1, x_2, \ldots, x_n) = n\) for \((x_1, x_2, \ldots, x_n) \in S\). The set of values is \(\{1, 2, 3, 4, 5, 6\}\).
- \(Y(x_1, x_2, \ldots, x_n) = \sum_{i=1}^n x_i\) for \((x_1, x_2, \ldots, x_n) \in S\). The set of possible values is \(\{0, 1, 2, 3, 4, 5, 6\}\).
- \(A = \{1, 11, 111, 1111, 11111, 111111\}\)
Run the simulation of the die-coin experiment 10 times. For each run, give the values of the random variables \(\bs{X}\), \(N\), and \(Y\) of the previous exercise. Count the number of times the event \(A\) occurs.
In the coin-die experiment , we have a coin and two distinct dice, say one red and one green. First the coin is tossed, and then if the result is heads the red die is thrown, while if the result is tails the green die is thrown. The coin score \(X\) and the score of the chosen die \(Y\) are recorded.
- Give the set of outcomes \(S\) in list form.
- Express \(X\) as a function on \( S \).
- Express \(Y\) as a function on \( S \).
- Give the event \(\{Y \ge 3\}\) as a subset of \( S \) in list form.
Answer
- \(\{(0,1), (0,2), (0,3), (0,4), (1,1), (1,2), (1,3), (1,4), (1,5), (1,6)\}\)
- \(X(i, j) = i\) for \((i, j) \in S\)
- \(Y(i, j) = j\) for \((i, j) \in S\)
- \(\{(0,3), (0,4), (1,3), (1,4), (1,5), (1,6)\}\)
Run the coin-die experiment 100 times, with various types of dice.
Sampling Models
Recall that many random experiments can be thought of as sampling experiments . For the general finite sampling model, we start with a population \(D\) with \(m\) (distinct) objects. We select a sample of \(n\) objects from the population. If the sampling is done in a random way, then we have a random experiment with the sample as the basic outcome. Thus, the set of outcomes of the experiment is literally the set of samples ; this is the historical origin of the term sample space . There are four common types of sampling from a finite population, based on the criteria of order and replacement . Recall the following facts from the section on combinatorial structures:
Samples of size \( n \) chosen from a population with \( m \) elements.
- If the sampling is with replacement and with regard to order, then the set of samples is the Cartesian power \(D^n\). The number of samples is \(m^n\).
- If the sampling is without replacement and with regard to order, then the set of samples is the set of all permutations of size \(n\) from \(D\). The number of samples is \(m^{(n)} = m (m - 1) \cdots [m - (n - 1)]\).
- If the sampling is without replacement and without regard to order, then the set of samples is the set of all combinations (or subsets ) of size \(n\) from \(D\). The number of samples is \(\binom{m}{n}\).
- If the sampling is with replacement and without regard to order, then the set of samples is the set of all multisets of size \(n\) from \(D\). The number of samples is \(\binom{m + n - 1}{n}\).
If we sample with replacement, the sample size \(n\) can be any positive integer. If we sample without replacement, the sample size cannot exceed the population size, so we must have \(n \in \{1, 2, \ldots, m\}\).
The basic coin and dice experiments are examples of sampling with replacement. If we toss a coin \(n\) times and record the sequence of scores (where as usual, 0 denotes tails and 1 denotes heads), then we generate an ordered sample of size \(n\) with replacement from the population \(\{0, 1\}\). If we throw \(n\) (distinct) standard dice and record the sequence of scores, then we generate an ordered sample of size \(n\) with replacement from the population \(\{1, 2, 3, 4, 5, 6\}\).
Suppose that the sampling is without replacement (the most common case). If we record the ordered sample \(\bs{X} = (X_1, X_2, \ldots, X_n)\), then the unordered sample \(\bs{W} = \{X_1, X_2, \ldots, X_n\}\) is a random variable (that is, a function of \(\bs{X}\)). On the other hand, if we just record the unordered sample \(\bs{W}\) in the first place, then we cannot recover the ordered sample. Note also that the number of ordered samples of size \(n\) is simply \(n!\) times the number of unordered samples of size \(n\). No such simple relationship exists when the sampling is with replacement. This will turn out to be an important point when we study probability models based on random samples, in the next section.
Consider a sample of size \(n = 3\) chosen without replacement from the population \(\{a, b, c, d, e\}\).
- Give \(T\), the set of unordered samples in list form.
- Give in list form the set of all ordered samples that correspond to the unordered sample \(\{b, c, e\}\).
- Note that for every unordered sample, there are 6 ordered samples.
- Give the cardinality of \(S\), the set of ordered samples.
Answer
- \(T = \left\{\{a,b,c\}, \{a,b,d\}, \{a,b,e\}, \{a,c,d\}, \{a,c,e\}, \{a,d,e\}, \{b,c,d\}, \{b,c,e\}, \{b,d,e\}, \{c,d,e\}\right\}\)
- \(\{(b,c,e), (b,e,c), (c,b,e), (c,e,b), (e,b,c), (e,c,b)\}\)
- 60
Traditionally in probability theory, an urn containing balls is often used as a metaphor for a finite population.
Suppose that an urn contains 50 (distinct) balls. A sample of 10 balls is selected from the urn. Find the number of samples in each of the following cases:
- Ordered samples with replacement
- Ordered samples without replacement
- Unordered samples without replacement
- Unordered samples with replacement
Answer
- \(97\,656\,250\,000\,000\,000\)
- \(37\,276\,043\,023\,296\,000\)
- \(10\,272\,278\,170\)
- \(62\,828\,356\,305\)
Suppose again that we have a population \(D\) with \(m\) (distinct) objects, but suppose now that each object is one of two types—either type 1 or type 0. Such populations are said to be dichotomous . Here are some specific examples:
- The population consists of persons, each either male or female .
- The population consists of voters, each either democrat or republican .
- The population consists of devices, each either good or defective .
- The population consists of balls, each either red or green .
Suppose that the population \(D\) has \(r\) type 1 objects and hence \(m - r\) type 0 objects. Of course, we must have \(r \in \{0, 1, \ldots, m\}\). Now suppose that we select a sample of size \(n\) without replacement from the population. Note that this model has three parameters: the population size \(m\), the number of type 1 objects in the population \(r\), and the sample size \(n\).
Let \(Y\) denote the number of type 1 objects in the sample. Then
- \(\#\{Y = k\} = \binom{n}{k} r^{(k)} (m - r)^{(n - k)}\) for each \(k \in \{0, 1, \ldots, n\}\), if the event is considered as a subset of \(S\), the set of ordered samples.
- \(\#\{Y = k\} = \binom{r}{k} \binom{m - r}{n - k}\) for each \(k \in \{0, 1, \ldots, n\}\), if the event is considered as a subset of \(T\), the set of unordered samples.
- The expression in (a) is \(n!\) times the expression in (b).
Proof
- \(\binom{n}{k}\) is the number of ways to pick the coordinates (in the ordered sample) where the type 1 objects will go, \(r^{(k)}\) is the number of ways to select a permutation of \(k\) type 1 objects, and \((m - r)^{(n-k)}\) is the number of ways to select a permutation of \(n - k\) type 0 objects. The result follows from the multiplication principle.
- \(\binom{r}{k}\) is the number of ways to select a combatination of \(k\) type 1 objects and \(\binom{m - r}{n - k}\) is the number of ways to select a combination of \(n - k\) type 0 objects. The result again follows from the multiplication principle.
- This result can be shown algebraically, but a combinatorial argument is better. For every combination of size \(n\) there are \(n!\) permutations of those objects.
A batch of 50 components consists of 40 good components and 10 defective components. A sample of 5 components is selected, without replacement. Let \(Y\) denote the number of defectives in the sample.
- Let \(S\) denote the set of ordered samples. Find \(\#(S)\).
- Let \(T\) denote the set of unordered samples. Find \(\#(T)\).
- As a subset of \(T\), find \(\#\{Y = k\}\) for each \(k \in \{0, 1, 2, 3, 4, 5\}\).
Answer
- \(254\,251\,200\)
- \(2\,118\,760\)
- \(\#\{Y = 0\} = 658\,008\), \(\#\{Y = 1\} = 913\,900\), \(\#\{Y = 2\} = 444\,600\), \(\#\{Y = 3\} = 93\,600\), \(\#\{Y = 4\} = 8\,400\), \(\#\{Y = 5\} = 252\)
Run the simulation of the ball and urn experiment 100 times for the parameter values in the last exercise: \( m = 50 \), \( r = 10 \), \( n = 5 \). Note the values of the random variable \(Y\).
Cards
Recall that a standard card deck can be modeled by the Cartesian product set \[D = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, j, q, k\} \times \{\clubsuit, \diamondsuit, \heartsuit, \spadesuit\}\] where the first coordinate encodes the denomination or kind (ace, 2–10, jack, queen, king) and where the second coordinate encodes the suit (clubs, diamonds, hearts, spades). Sometimes we represent a card as a string rather than an ordered pair (for example \(q \heartsuit\) rather than \((q, \heartsuit)\) for the queen of hearts).
Most card games involve sampling without replacement from the deck \(D\), which plays the role of the population. Thus, the basic card experiment consists of dealing \(n\) cards from a standard deck without replacement; in this special context, the sample of cards is often referred to as a hand . Just as in the general sampling model, if we record the ordered hand \(\bs{X} = (X_1, X_2, \ldots, X_n)\), then the unordered hand \(\bs{W} = \{X_1, X_2, \ldots, X_n\}\) is a random variable (that is, a function of \(\bs{X}\)). On the other hand, if we just record the unordered hand \(\bs{W}\) in the first place, then we cannot recover the ordered hand. Finally, recall that \(n = 5\) is the poker experiment and \(n = 13\) is the bridge experiment . The game of poker is treated in more detail in the chapter on Games of Chance.
Suppose that a single card is dealt from a standard deck. Let \(Q\) denote the event that the card is a queen and \(H\) the event that the card is a heart. Give each of the following events in list form:
- \(Q\)
- \(H\)
- \(Q \cup H\)
- \(Q \cap H\)
- \(Q \setminus H\)
Answer
- \(Q = \{q \clubsuit, q \diamondsuit, q \heartsuit, q \spadesuit\}\)
- \(H = \{1 \heartsuit, 2 \heartsuit, \ldots, 10 \heartsuit, j \heartsuit, q \heartsuit, k \heartsuit\}\)
- \(Q \cup H = \{1 \heartsuit, 2 \heartsuit, \ldots, 10 \heartsuit, j \heartsuit, q \heartsuit, k \heartsuit, q \clubsuit, q \diamondsuit, q \spadesuit\}\)
- \(Q \cap H = \{q \heartsuit\}\)
- \(Q \setminus H = \{q \clubsuit, q \diamondsuit, q \spadesuit\}\)
In the card experiment, set \(n = 1\). Run the experiment 100 times and count the number of times each event in the previous exercise occurs.
Suppose that two cards are dealt from a standard deck and the sequence of cards recorded. Let \( S \) denote the set of outcomes, and let \(Q_i\) denote the event that the \(i\)th card is a queen and \(H_i\) the event that the \(i\)th card is a heart for \(i \in \{1, 2\}\). Find the number of outcomes in each of the following events:
- \(S\)
- \(H_1\)
- \(H_2\)
- \(H_1 \cap H_2\)
- \(Q_1 \cap H_1\)
- \(Q_1 \cap H_2\)
- \(H_1 \cup H_2\)
Answer
- 2652
- 663
- 663
- 156
- 51
- 51
- 1170
Consider the general card experiment in which \(n\) cards are dealt from a standard deck, and the ordered hand \(\bs{X}\) is recorded.
- Give cardinality of \(S\), the set of values of the ordered hand \( \bs{X} \).
- Give the cardinality of \(T\), the set of values of the unordered hand \(\bs{W}\).
- How many ordered hands correspond to a given unordered hand?
- Explicitly compute the numbers in (a) and (b) when \(n = 5\) (poker).
- Explicitly compute the numbers in (a) and (b) when \(n = 13\) (bridge).
Answer
- \(\#(S) = 52^{(n)}\)
- \(\#(T) = \binom{52}{n}\)
- \(n!\)
- \(311\,875\,200\), \(2\,598\,960\)
- \(3\,954\,242\,643\,911\,239\,680\,000\), \(635\,013\,559\,600\)
Consider the bridge experiment of dealing 13 cards from a deck and recording the unordered hand. In the most common point counting system , an ace is worth 4 points, a king 3 points, a queen 2 points, and a jack 1 point. The other cards are worth 0 points. Let \( S \) denote the set of outcomes of the experiment and \(V\) the point value of the hand.
- Find the set of possible values of \(V\).
- Find the cardinality of the event \(\{V = 0\}\) as a subset of \( S \).
Answer
- \(\{0, 1, \ldots, 37\}\)
- \(\#\{V = 0\} = 2\,310\,789\,600\)
In the card experiment, set \(n = 13\) and run the experiment 100 times. For each run, compute the value of each of the random variable \(V\) in the previous exercise.
Consider the poker experiment of dealing 5 cards from a deck. Find the cardinality of each of the events below, as a subset of the set of unordered hands.
- \(A\): the event that the hand is a full house (3 cards of one kind and 2 of another kind).
- \(B\): the event that the hand has 4 of a kind (4 cards of one kind and 1 of another kind).
- \(C\): the event that all cards in the hand are in the same suit (the hand is a flush or a straight flush ).
Answer
- \(\#(A) = 3744\)
- \(\#(B) = 624\)
- \(\#(C) = 5148\)
Run the poker experiment 1000 times. Note the number of times that the events \(A\), \(B\), and \(C\) in the previous exercise occurred.
Consider the bridge experiment of dealing 13 cards from a standard deck. Let \( S \) denote the set of unordered hands, \(Y\) the number of hearts in the hand, and \(Z\) the number of queens in the hand.
- Find the cardinality of the event \(\{Y = y\}\) as a subset of \( S \) for each \(y \in \{0, 1, \ldots, 13\}\).
- Find the cardinality of the event \(\{Z = z\}\) as a subset of \( S \) for each \(z \in \{0, 1, 2, 3, 4\}\).
Answer
- \(\#(Y = y) = \binom{13}{y} \binom{39}{13 - y}\) for \(y \in \{0, 1, \ldots, 13\}\)
- \(\#(Z = z) = \binom{4}{z} \binom{48}{4 - z}\) for \(z \in \{0, 1, 2, 3, 4\}\)
Geometric Models
In the experiments that we have considered so far, the sample spaces have all been discrete (so that the set of outcomes is finite or countably infinite). In this subsection, we consider Euclidean sample spaces where the set of outcomes \(S\) is continuous in a sense that we will make clear later. The experiments we consider are sometimes referred to as geometric models because they involve selecting a point at random from a Euclidean set.
We first consider Buffon's coin experiment , which consists of tossing a coin with radius \(r \le \frac{1}{2}\) randomly on a floor covered with square tiles of side length 1. The coordinates \((X, Y)\) of the center of the coin are recorded relative to axes through the center of the square in which the coin lands. Buffon's experiments are studied in more detail in the chapter on Geometric Models and are named for Compte de Buffon
In Buffon's coin experiment, let \( S \) denote the set of outcomes, \(A\) the event that the coin does not touch the sides of the square, and let \(Z\) denote the distance form the center of the coin to the center of the square.
- Describe \(S\) as a Cartesian product.
- Describe \(A\) as a subset of \(S\).
- Describe \(A^c\) as a subset of \(S\).
- Express \(Z\) as a function on \(S\).
- Express the event \(\{X \lt Y\}\) as a subset of \(S\).
- Express the event \(\left\{Z \leq \frac{1}{2}\right\}\) as a subset of \(S\).
Answer
- \(S = \left[-\frac{1}{2}, \frac{1}{2}\right]^2\)
- \(A = \left[r - \frac{1}{2}, \frac{1}{2} - r\right]^2\)
- \(A^c = \left\{(x, y) \in S: x \lt r - \frac{1}{2} \text{ or } x \gt \frac{1}{2} - r \text{ or } y \lt r - \frac{1}{2} \text{ or } y \gt \frac{1}{2} - r\right\}\)
- \(Z(x, y) = \sqrt{x^2 + y^2}\) for \((x, y) \in S\)
- \(\{X \lt Y\} = \{(x, y) \in S: x \lt y\}\)
- \(\{Z \lt \frac{1}{2}\} = \left\{(x, y) \in S: x^2 + y^2 \lt \frac{1}{4}\right\}\)
Run Buffon's coin experiment 100 times with \(r = 0.2\). For each run, note whether event \(A\) occurs and compute the value of random variable \(Z\).
A point \((X, Y)\) is chosen at random in the circular region of radius 1 in \(\R^2\) centered at the origin. Let \(S\) denote the set of outcomes. Let \(A\) denote the event that the point is in the inscribed square region centered at the origin, with sides parallel to the coordinate axes. Let \(B\) denote the event that the point is in the inscribed square with vertices \((\pm 1, 0)\), \((0, \pm 1)\).
- Describe \(S\) mathematically and sketch the set.
- Describe \(A\) mathematically and sketch the set.
- Describe \(B\) mathematically and sketch the set.
- Sketch \(A \cup B\)
- Sketch \(A \cap B\)
- Sketch \(A \cap B^c\)
Answer
- \(S = \left\{(x, y): x^2 + y^2 \le 1\right\}\)
- \(A = \left\{(x, y): -\frac{1}{\sqrt{2}} \le x \le \frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}} \le y \le \frac{1}{\sqrt{2}}\right\}\)
- \(B = \left\{(x, y) \in S: -1 \le \left|x + y\right| \le 1, -1 \le \left|y - x\right| \le 1\right\}\)
Reliability
In the simple model of structural reliability , a system is composed of \(n\) components, each of which is either working or failed . The state of component \(i\) is an indicator random variable \(X_i\), where 1 means working and 0 means failure. Thus, \(\bs{X} = (X_1, X_2, \ldots, X_n)\) is a vector of indicator random variables that specifies the states of all of the components, and therefore the set of outcomes of the experiment is \(S = \{0, 1\}^n\). The system as a whole is also either working or failed, depending only on the states of the components and how the components are connected together. Thus, the state of the system is also an indicator random variable and is a function of \( \bs X \). The state of the system (working or failed) as a function of the states of the components is the structure function .
A series system is working if and only if each component is working. The state of the system is \[U = X_1 X_2 \cdots X_n = \min\left\{X_1, X_2, \ldots, X_n\right\}\]
A parallel system is working if and only if at least one component is working. The state of the system is \[V = 1 - \left(1 - X_1\right)\left(1 - X_2\right) \cdots \left(1 - X_n\right) = \max\left\{X_1, X_2, \ldots, X_n\right\}\]
More generally, a \(k\) out of \(n\) system is working if and only if at least \(k\) of the \(n\) components are working. Note that a parallel system is a 1 out of \(n\) system and a series system is an \(n\) out of \(n\) system. A \(k\) out of \(2 k\) system is a majority rules system .
The state of the \(k\) out of \(n\) system is \( U_{n,k} = \bs 1\left(\sum_{i=1}^n X_i \ge k\right) \). The structure function can also be expressed as a polynomial in the variables.
Explicitly give the state of the \(k\) out of 3 system, as a polynomial function of the component states \((X_1, X_2, X_3)\), for each \(k \in \{1, 2, 3\}\).
Answer
- \(U_{3,1} = X_1 + X_2 + X_3 - X_1 X_2 - X_1 X_3 - X_2 X_3 + X_1 X_2 X_3\)
- \(U_{3,2} = X_1 X_2 + X_1 X_3 + X_2 X_3 - 2 \, X_1 X_2 X_3\)
- \(U_{3,3} = X_1 X_2 X_3\)
In some cases, the system can be represented as a graph or network . The edges represent the components and the vertices the connections between the components. The system functions if and only if there is a working path between two designated vertices, which we will denote by \(a\) and \(b\).
Find the state of the Wheatstone bridge network shown below, as a function of the component states. The network is named for Charles Wheatstone.
Answer
Not every function \(u: \{0, 1\}^n \to \{0, 1\}\) makes sense as a structure function. Explain why the following properties might be desirable:
- \(u(0, 0, \ldots, 0) = 0\) and \(s(1, 1, \ldots, 1) = 1\)
- \(u\) is an increasing function, where \(\{0, 1\}\) is given the ordinary order and \(\{0, 1\}^n\) the corresponding product order.
- For each \(i \in \{1, 2, \ldots, n\}\), there exist \(\bs{x}\) and \(\bs{y}\) in \(\{0, 1\}^n\) all of whose coordinates agree, except \(x_i = 0\) and \(y_i = 1\), and \(u(\bs{x}) = 0\) while \(u(\bs{y}) = 1\).
Answer
- This means that if all components have failed then the system has failed, and if all components are working then the system is working.
- This means that if a particular component is changed from failed to working, then the system may also go from failed to working, but not from working to failed. That is, the system can only improve.
- This means that every component is relevant to the system, that is, there exists a configuration in which changing component \(i\) from failed to working changes the system from failed to working.
The model just discussed is a static model . We can extend it to a dynamic model by assuming that component \(i\) is initially working, but has a random time to failure \(T_i\), taking values in \([0, \infty)\), for each \(i \in \{1, 2, \ldots, n\}\). Thus, the basic outcome of the experiment is the random vector of failure times \((T_1, T_2, \ldots, T_n)\), and so the set of outcomes is \([0, \infty)^n\).
Consider the dynamic reliability model for a system with structure function \(u\) (valid in the sense of the previous exercise).
- The state of component \(i\) at time \(t \ge 0\) is \(X_i(t) = \bs{1}\left(T_i \gt t\right)\).
- The state of the system at time \(t\) is \(X(t) = s\left[X_1(t), X_2(t), \ldots, X_n(t)\right]\).
- The time to failure of the system is \(T = \min\{t \ge 0: X(t) = 0\}\).
Suppose that we have two devices and that we record \((X, Y)\), where \(X\) is the failure time of device 1 and \(Y\) is the failure time of device 2. Both variables take values in the interval \([0, \infty)\), where the units are in hundreds of hours. Sketch each of the following events:
- The set of outcomes \(S\)
- \(\{X \lt Y\}\)
- \(\{X + Y \gt 2\}\)
Answer
- \( S = [0, \infty)^2 \), the first quadrant of the coordinate plane.
- \( \{X \lt Y\} = \{(x, y) \in S: x \lt y\} \). This is the region below the diagonal line \( x = y \).
- \( \{X + Y \gt 2\} = \{(x, y) \in S: x + y \gt 2 \). This is the region above (or to the right) of the line \( x + y = 2 \).
Genetics
Please refer to the discussion of genetics in the section on random experiments if you need to review some of the definitions in this section.
Recall first that the ABO blood type in humans is determined by three alleles: \(a\), \(b\), and \(o\). Furthermore, \(o\) is recessive and \(a\) and \(b\) are co-dominant.
Suppose that a person is chosen at random and his genotype is recorded. Give each of the following in list form.
- The set of outcomes S
- The event that the person is type \(A\)
- The event that the person is type \(B\)
- The event that the person is type \(AB\)
- The event that the person is type \(O\)
Answer
- \(S = \{aa, ab, ao, bb, bo, oo\}\)
- \(A = \{aa, ao\}\)
- \(B = \{bb, bo\}\)
- \(AB = \{ab\}\)
- \(O = \{oo\}\)
Suppose next that pod color in certain type of pea plant is determined by a gene with two alleles: \(g\) for green and \(y\) for yellow, and that \(g\) is dominant.
Suppose that \(n\) (distinct) pea plants are collected and the sequence of pod color genotypes is recorded.
- Give the set of outcomes \(S\) in Cartesian product form and find \(\#(S)\).
- Let \(N\) denote the number of plants with green pods. Find \(\#(N = k)\) (as a subset of \( S \)) for each \(k \in \{0, 1, \ldots, n\}\).
Answer
- \(S = \{gg, gy, yy\}^n \), \(\#(S) = 3^n\)
- \(\binom{n}{k} 2^k\)
Next consider a sex-linked hereditary disorder in humans (such as colorblindness or hemophilia). Let \(h\) denote the healthy allele and \(d\) the defective allele for the gene linked to the disorder. Recall that \(d\) is recessive for women.
Suppose that \(n\) women are sampled and the sequence of genotypes is recorded.
- Give the set of outcomes \(S\) in Cartesian product form and find \(\#(S)\).
- Let \(N\) denote the number of women who are completely healthy (genotype \(hh\)). Find \(\#(N = k)\) (as a subset of \( S \)) for each \(k \in \{0, 1, \ldots, n\}\).
Answer
- \(S = \{hh, hd, dd\}^n\), \(\#(S) = 3^n\)
- \(\binom{n}{k} 2^{n-k}\)
Radioactive Emissions
The emission of elementary particles from a sample of radioactive material occurs in a random way. Suppose that the time of emission of the \(i\)th particle is a random variable \(T_i\) taking values in \((0, \infty)\). If we measure these arrival times , then basic outcome vector is \((T_1, T_2, \ldots)\) and so the set of outcomes is \(S = \{(t_1, t_2, \ldots): 0 \lt t_1 \lt t_2 \lt \cdots\}\).
Run the simulation of the gamma experiment in single-step mode for different values of the parameters. Observe the arrival times.
Now let \(N_t\) denote the number of emissions in the interval \((0, t]\). Then
- \(N_t = \max\left\{n \in \N_+: T_n \le t\right\}\).
- \(N_t \ge n\) if and only if \(T_n \le t\).
Run the simulation of the Poisson experiment in single-step mode for different parameter values. Observe the arrivals in the specified time interval.
Statistical Experiments
In the basic cicada experiment, a cicada in the Middle Tennessee area is captured and the following measurements recorded: body weight (in grams), wing length, wing width, and body length (in millimeters), species type, and gender. The cicada data set gives the results of 104 repetitions of this experiment.
- Define the set of outcomes \( S \) for the basic experiment.
- Let \(F\) be the event that a cicada is female. Describe \(F\) as a subset of \( S \). Determine whether \(F\) occurs for each cicada in the data set.
- Let \(V\) denote the ratio of wing length to wing width. Compute \(V\) for each cicada.
- Give the set of outcomes for the compound experiment that consists of 104 repetitions of the basic experiment.
Answer
For gender, let 0 denote female and 1 male, for species, let 1 denote tredecula, 2 tredecim, and 3 tredecassini.
- \(S = (0, \infty)^4 \times \{0, 1\} \times \{1, 2, 3\}\)
- \(F = \{(x_1, x_2, x_3, x_4, y, z) \in S: y = 0\}\)
- \(S^{104}\)
In the basic M&M experiment, a bag of M&Ms (of a specified size) is purchased and the following measurements recorded: the number of red, green, blue, yellow, orange, and brown candies, and the net weight (in grams). The M&M data set gives the results of 30 repetitions of this experiment.
- Define the set of outcomes \( S \) for the basic experiment.
- Let \(A\) be the event that a bag contains at least 57 candies. Describe \(A\) as a subset of \( S \).
- Determine whether \(A\) occurs for each bag in the data set.
- Let \(N\) denote the total number of candies. Compute \(N\) for each bag in the data set.
- Give the set of outcomes for the compound experiment that consists of 30 repetitions of the basic experiment.
Answer
- \(S = \N^6 \times (0, \infty)\)
- \(A = \{(n_1, n_2, n_3, n_4, n_5, n_6, w) \in S: n_1 + n_2 + \cdots + n_6 \gt 57\}\)
- \(S^{30}\)


<IP_ADDRESS>: Capitalization
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Some teachers blame text messaging. After all, how many times have you written out a quick text message, not worrying about where you need to capitalize? You, like many others, have probably done this quite often.
Interestingly, capitalization errors appeared on the most-common error list even before everyone was walking around with their heads over their cell phones text messaging every other minute.
Of course, the reality is that teachers now often complain that students don’t even capitalize "I" in a sentence. That could likely be related to text messaging. Still, other capitalization errors may not be.
Whatever the reasons behind the errors, it’s important to be aware of capitalization rules. Be careful to capitalize words that should be capitalized, such as proper nouns, and not capitalize words that should not be capitalized.
But, how can you know what should and shouldn’t be capitalized? If you are unsure, review the differences between proper nouns and common nouns in the Parts of Speech chapter.
Capitalization errors can really have a negative impact on the impressions your readers have of your writing. Take a look at the following sentence:
Now, take a look at the sentence with correct capitalization:
It’s important to edit carefully and review the rules if necessary!