Section 1.3: Promoting Inclusion Through Storytelling and Dialogue
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- Amber E. Kelly & Kathleen E. McAuliff
- Rebus Community
The chapter will share insights on this innovative strategy for promoting inclusion among individuals with a disability using community dialogue and events.
The Big Picture
According to the World Health Organization (WHO), more than 1 billion people live w ith a disability (WHO, 2017). Globally and nationally, disabilities are more commonly found in areas of poverty. Approximately 26% of people in the U.S. have a disability, with the most common disabilities being related to mobility or cognition (Okoro et al., 2018). As people live longer, they are more likely to experience a disability (Bialik, 2020). Although there is a higher prevalence of disability than some might guess, individuals with a disability still experience adverse outcomes when compared to individuals who do not have a disability, including reduced access to healthcare, educational, and employment opportunities (Okoro et al., 2018). While the causes of these adverse outcomes are multi-faceted, one potential avenue for positive social change is increasing visibility and educating the community about the experiences of people with disabilities. While overall perceptions are changing, there are still many myths about disabilities. One of the most common myths is that all disabilities and chronic conditions are visible.
In fact, 96% of people with a chronic condition do not have a condition that is visible, and 73% of people with a severe disability do not use a personal assistive device such as a wheelchair or walker ( Disabled World, 2020 ) . Educating people through first-hand experiences can help dispel these myths, and illuminate ways in which the community could change to promote inclusion for people with disabilities. To move our theoretical framework of awareness and storytelling to practice, we sought funding and begin to put the project together. See highlights below of statistics regarding disabilities:
World Health Organization (WHO)
Disability and Health, 1 December 2020
Key Facts
The number of people with disability are dramatically increasing. This is due to demographic trends and increases in chronic health conditions, among other causes.
- Over 1 billion people live with some form of disability.
- Almost everyone is likely to experience some form of disability - temporary or permanent - at some point in life.
- People with disbaility are disproportionately affected during the COVID-19 pandemic
- If health services for people with disability exist, they are invariably of poor qualty or under resourced.
- There is an urgent need to scale up disability services in primary healthcare, particularly in rehabilitation interventions.
The goal of Dare2Dialogue was to bring awareness and engage in storytelling around challenging topics to promote change, specifically focused on individuals with a disability as a means to remove the negative stigma that can be associated with a disability. This goal was initially achieved by having two individuals who live with disabilities share their stories to highlight challenges and encourage dialogue among those who may have the privilege of not living with a disability. Subsequently, three community discussions (including one documentary screening) created an opportunity for 96 Dare2Dialogue attendees to challenge their thinking around inclusion. The chapter will share insights on this innovative strategy for promoting inclusion for individuals with a disability among community events.
Community Needs
Traditionally, local organizations that serve individuals with a physical or cognitive impairment ensure that events prioritize ability. Some examples used in hosting events include using locations that are wheelchair accessible and having sitting space for caregivers. Organizations share a challenge of attracting others to attend that typically do not attend ability-focused events. Attendees are ability advocates, and there is an interest in broadening the audience’s diversity when it pertains to interest in ability-related issues. One local initiative emphasizes inclusion through the medium of film ( Over The Rhine International Film Festival, 2021 ). With an open door to help close this gap in initiatives that focus on living with a disability, there were opportunities to create additional events. Further, People’s Liberty was a philanthropic lab that awarded 120 visionaries grants to impact their community in an innovative way. The five-year initiative focused on empowering individuals to help improve communities’ well-being where they lived and thrived. At the 2.5-year mark of this initiative, People’s Liberty hosted a midway celebration, “Intermission”, which allowed for twenty grantees to host an innovative event in a storefront space. The first event using this storefront space was the pilot for Dare2Dialogue, and it was the only event during the series that focused on the inclusion of individuals living with a disability. It was one of the most attended events during the celebration, which prompted the development of additional events in ability.
Collaborative Partners
When selecting collaborative partners, they should share the collective vision for promoting the inclusion of individuals living with a disability. Community Engagement Collective (CEC) partnered with Starfire , an organization focusing on empowering leaders to build community and inclusion alongside people with developmental disabilities and People’s Liberty using their storefront building as the venue for all Dare2Dialogue events.
Language Matters
Using language that does not demean a population is critical to consider for working within community settings. The disability field has evolved in terms of the preferred language and continues to grow. The American Psychological Association (2019) recommends using language to promote solidarity, respect, and ultimately honor all individuals as human beings. First-person language focuses on the individual, while individual first language emphasizes the disability. Dunn and Andrews (2015) make a case for using both first-person and identity-first language when doing work in the disability field and taking a flexible approach is respectful. Ultimately, using the term preferred by the individuals with the lived experience is the best option (American Psychological Association, 2019).
What Does Inclusion of Individuals with a Disability Really Mean?
Inclusion ensures that people with disabilities have their voices and experiences heard by people who may not see the same barriers or opportunities that exist in a community. If only people without disabilities are making decisions, certain barriers and opportunities may not be identified, thus excluding vital community members. For example, a community may not offer physical public spaces which are ADA (Americans with Disabilities Act) compliant, or a library may not offer spaces for people with sensory processing disorders, such as autism spectrum disorder. As no two people without disabilities are the same, people with disabilities are not a monolith, even if they have the same type of disability. Different disabilities bring different strengths and challenges, so creating an inclusive environment where all voices can be respected and heard is critical. Because most people do not have a disability, openness to learning and listening is imperative to make our communities truly inclusive and create larger community change. Storytelling in diverse settings allows speakers to share their stories and make connections with others that may not have previously had the opportunity.
Storytelling and Inclusion
One method of including individuals with disabilities is through storytelling. Storytelling allows people with disabilities to share their experiences in a structured format where participants actively listen to another person’s experiences. One of the 10 principles, that is key to intergroup relations is to provide opportunities for members of different groups to get to know one another as individuals (ASDC for the Community Foundations/Intergroup Relations Program, 1999) . Through listening to individual stories from people with disabilities, people without disabilities can put a face to an experience, rather than stereotyping. Additionally, it also provides an opportunity for people to build a sense of community with one another. Storytelling is a unique type of participatory action research method that empowers a participant to shape and share their own narrative, communicating the truth and emotional impact of their experiences (Bailey & Tilley, 2002). Without using prompts or being guided by a researcher’s agenda, participants with disabilities can decide which event(s) and experiences they can share.
Community conversations are another method of bringing together people with disabilities and people without disabilities (Carter & Bumble, 2018). Community conversations use an asset-based focus, solution-focused framing, awareness building, and a shared commitment to improving the community for individuals with disabilities. While storytelling and community conversations are similar, storytelling was selected for this event so that the focus could be on the speakers with disabilities sharing their experiences and shaping the narrative to promote awareness, so that the community members could listen, reflect, and, for some, redefine the ways they perceived someone with a disability.
Dare2Dialogue’s Linkage to Community Psychology Practice
Empowering individuals with disabilities through creating spaces where they can shape the narrative and a community can listen is both an act of social justice and empowerment.
Dare2Dialogue fused together community conversation and storytelling (Dare2Dialogue, 2021) through the use of multiple principles of community psychology. Community psychology explicitly states a sense of community, respect for human diversity, social justice, and empowerment/citizen participation as some of its core values (Prilleltensky & Nelson, 1997). This particular event is focused on fostering a sense of community through including and listening to the experiences of individuals with disabilities. As inclusivity is often lacking for people with disabilities, creating a sense of community where everyone feels like they belong requires community members to understand the ways in which they (individually, interpersonally, or as a community) may be excluding community members who have disabilities. Furthermore, as disability is a type of diversity, it is also important that people with disabilities are represented and respected in the community. As mentioned earlier in this case story, individuals with disabilities often experience adverse outcomes because of social injustices, such as reduced access to employment, education, and discriminatory practices.
Description of the Project
Dare2Dialogue Series on Inclusion consisted of events where an individual (or individuals) with a disability shared their experiences with the larger community, including people without disabilities, in order to educate, enlighten, and promote dialogue about community inclusion for people with disabilities. The series included three distinct events which are: (1) storytelling from the perspective of someone living with a physical disability , (2) storytelling from the perspective of living with a cognitive disability, and (3) movie screening sharing the lives of individuals with a disability.
For recruitment, a snowball sampling strategy was used (Noy, 2008). A snowball sampling strategy is commonly used in qualitative research and involves one participant using their network (i.e., people they know) to recruit additional participants. It is particularly useful in populations that may be difficult to reach or recruit. In some cases, a researcher will ask the participant to provide the contact information for additional potential participants to facilitate recruitment. In other cases, the initial participant(s) will recruit the potential participants, as they already have an established relationship. This event used the latter method for recruiting participants. Each event took place in the same venue, located in downtown Cincinnati, Ohio, close to public transportation, and was accessible. The first event took place during lunchtime, and the second and third events took place in the evening. When arriving at the first two events, attendees received a card to complete a word cloud . For word clouds, the statement should allow for a one-word response only. The same prompt was used. Below is a picture of a cloud with the prompt in it which says, “in one word what comes to mind when you think about inclusiveness?
Attendees had the option of texting their response, completing via website link, or sharing their reactions with a volunteer. A Quick Response (QR) code on the card helped to make the connection to the link easier. The word cloud was shown on a projector for attendees to view. Before the community member spoke, the guests enjoyed free food and drinks since it was close to mealtimes. Attendees took the time to mix and mingle before each event to facilitate community building.
Physical Ability
Word Cloud from 1st Event
Image above is a “word cloud” or collage of words such as fairness, access, opportunity, ability…
Narrative One
Speech from 1st Event
Hello everyone, I would like to start by thanking everyone for showing up for our discussion on inclusion in a midwestern city. So for starters, what is Inclusion? According to Webster, it’s the act of or state of being included. That’s it, it’s just that simple. Now add people to inclusion and it becomes much more convoluted because every one of us is different. Just take a look around the room and you can see the diversity here. Every one of us has multiple qualities that make us unique, be it our physical differences such as our height, color of our skin, body shape, hair, you name it. But we also have different nonphysical qualities which aren’t so prevalent, such as our upbringing, education, and financial backgrounds, the way we process information, biases, etc. There is no cookie-cutter form of inclusion that will work for everyone so inclusion will look different in different settings. For example, at my job, I am the only person there who doesn’t have to walk anywhere. Steps… pssh I’m not walking up steps. I have my own way of getting around and it doesn’t cause me to burn much energy. Just look at this bad boy. 2 all-terrain front wheels, good for climbing obstacles, fog headlights, padded leather back and armrests, and a hydraulic system that’d put most low-riders to shame. It doesn’t get any better than this.
“This is an awesome chair but I haven’t always been this blessed, over 9 years ago I used to have to walk like you all until one morning as I was riding my bicycle to my job at the hospital and was hit by a guy making a left turn who didn’t see me”.
 So when they tested to see if I could breathe on my own, I’d do it until I couldn’t take it anymore. Each time I’d go a little longer. I relearned how to eat without choking because I was on a feeding tube for 3 months.   Eventually, I learned to feed myself, and I was able to strengthen my muscles enough to go home.
All of these tasks were difficult but I would prefer them over not being able to see my daughters for 3 months. You see, my accident happened during “flu season” at the hospitals, and during that time children aren’t allowed past a certain point. Thankfully, I had my family who brought me pictures of them. It wasn’t the same as seeing them in person but it was better than nothing. My family helped tremendously with my recovery so when I think of inclusion, I think of FAMILY because to be included in a group sometimes can be like gaining a new family.
After my accident, I saw my city differently. I became aware of how inaccessible it really was. I have come across sidewalks that I cannot access so I had to ride in the street with busy traffic, businesses without a ramp or elevator. Transportation for wheelchair users is inconvenient. I began thinking of ways to improve the city in that arena, but I didn’t have to think I could change anything alone nor did I have the resources, nor did I know where to start because all of this was still new to me, so my ideas were placed on the back burner. It wasn’t ‘til I interviewed at my job that I started back thinking about Inclusion. Because when I went for my interview, I needed help getting inside because there weren’t any automated doors. My now Supervisor asked if I saw any obstacles for me to work there and the doors were my only issue. Within a few months, plans were in place for an automated door to be put in and now it is truly accessible all because of the pebbles I threw in the pond. So I’ve learned that little things can cause big ripples and that my city is ready for change.
Cognitive Ability
Narrative Two
Speech from 2nd Event
I was born 3 months premature. Doctors thought I would not be able to walk (paralyzed). I weighed 1lb. Mom said she could fit me in the palm of her hand). My disability is called cerebral palsy. I have 1 older sister and 2 younger brothers. It is hard being the oldest brother with a disability because I thought I would be the first one to do everything like drive. I do not like asking people to do stuff for me because I feel like I can do it all on my own. But we all need help sometimes.
Elementary School
My mom asked me did I want to go to a regular school or a school for people with disabilities. My answer was a regular school. I wanted to go to a regular school because I felt like I had to do it. I had a choice. Everyone does not have a choice. Sometimes people make the choice for you. When I came home from school one day, I asked my sister “Why am I different?” She did not have a response to my question. Her mind was puzzled. I asked her because I wanted to know what her response would be. Later, I was bullied in the 1st grade. A student at school picked on me. I stood up for myself. It felt good. Today, I do not like seeing people getting picked on.
I played football as a kid. I played football ages 7-12 (except age 9). It was important for my dad to coach me so I could feel like a regular kid. I loved having my dad coach me because he understood me. I loved playing football. When I was 10 years old, my 7-year-old brother played in a game with me. I remember I asked my brother did he want to fill in because we did not have enough players to play that game. He said “Yeah.” I said, “These are 10-year-olds, you are 7, are you sure you want to play?” I asked him twice. He said “Yeah.” He held his own and he made tackles. I was proud of my little brother. At 7 years old, I would fall to the ground so I wouldn’t get tackled. I played basketball too and my dad coached me as well. I was a better basketball player than a football player.
Gym Class Story
This one particular afternoon in school, the gym teacher had asked me to jump rope. My response was no because I did not feel comfortable jumping rope in front of the whole class. I felt embarrassed to be put on the spot like that because she knew of my circumstances. I was upset and that was the hardest thing I went through as a child. It felt great to have my classmates have my back. They asked the teacher why she had me jump rope in front of the whole class when she knew of my circumstance. She did not have a response.
Junior High School
When I first went to Junior High School, I was nervous because it was a new environment. I did not know how I would adapt to the new place. Being accepted in the National Honor Society was an excellent thing. It was great to realize that I was doing good in school. However, in the 7th grade, there was a teacher who did not want to modify the work for me in my social studies class. To avoid the problem, I just went to a special education social studies class instead. I was disappointed because I wanted to be in a regular social studies class with all of my friends.
High School
I had a wonderful person that scribed for me and helped me with my thoughts. She worked with me since junior high school. The gym teacher was nice. She was very understanding and helpful. My senior year was the most exciting out of my 4 years of high school because I got to participate on the football team. I wanted to motivate people and be an encourager. I did not let anything stop me from doing what I wanted to do. Graduation was emotional. I thought about all the things from elementary school from when teachers did not want to modify my work to graduating high school. I felt like I had completed my goal to graduate from high school.
StarfireU
I wanted to do StarfireU because it was exciting and a 4-year program. I was the youngest in the program. It helped me to learn how to communicate with others. It helped me realize how to be a good friend to people. I learned how to make sure everybody felt like they were included. I remember a few times in the building some people would ask me to pray for them. I would take them to a room and pray for them. This is how I found my calling.
My Calling
My calling is to preach in ministry. It was hard at first to accept the calling because I did not know if people would accept me because of my disability. There were a lot of sleepless nights struggling to accept my calling. Once I accepted my calling, it felt great. The goal of my sermons is to encourage people to be great. I want to let people know they should live life to the fullest and be great at what they do.
Movie Screening
For the final event, a movie screening, which was a collaboration between two local nonprofit organizations Starfire and Community Engagement Collective took place. Partnering organizations shared the event on their social media platforms. Attendees enjoyed fresh popcorn and pizza before the movie. The previous event used in-person storytelling as the foundation; however, this event used movies as a form of art. The movie Wretches and Jabberers follows protagonists Tracy and Larry as they travel the world sharing their advocacy message for individuals with autism. After the movie, attendees discussed their reactions to the film in small groups. Some of the group sharing included statements like “Having my voice heard is life-changing,” “More like you than not,” and ” The humanity of acceptance.” (See Figure 3)
Figure 3 Illustration from Movie Screening Dialogue
Outcomes and Impact
The goal of Dare2Dialogue was to bring awareness and engage in dialogue around challenging topics to promote change (See Figure 4). The purpose of the community discussions on inclusion was to bring awareness to the lack of inclusion of individuals with a disability using storytelling, community conversations, and movie screening. After the launch of the Dare2Dialogue inclusion series, the community partners collaborated on various initiatives including training, speaking engagements, and conference planning. The common thread of these initiatives focused on the inclusion of individuals living with a disability. Figure 4 below shows that in the program, there were 96 attendees, 3 events for the purpose of bringing awareness to the importance of inclusion.
Figure 4: Impact of Dare2Dialogue Inclusion Series
Lessons Learned
We learned through this process three important factors: (1) Do not use identity-first language (e.g., disability), (2) partner with other organizations for hosting events, and (3) engage audience members in multiple ways.
Hosting three different events gave insight into how to plan future events that emphasize individuals of diverse abilities. When striving to foster inclusion within events, it is important not to label the event with disability or ability. This limits the diversity of audience members in attendance. In turn, the conversation’s potential impact is limited because the audience members tend to understand the issues within the ability space. People are creatures of habit and tend to participate in events that address a topic of interest versus an unfamiliar issue. For instance, ability advocates are more likely to attend events that focus on ability. To target non-advocates, use terminology that describes the event’s activities, not the storytellers and or community members. For example, if a storyteller is living with autism and is sharing gardening, emphasize gardening in marketing materials. In this example, the audience members will build connections due to the collective interest in gardening, not autism. Example words can include diversity, inclusion, conversation, storytelling, dialogue, and community connections in event marketing materials. Intentional marketing helps to reach a broad audience that may have been less likely to connect otherwise.
Partnering with organizations for event planning and implementation can achieve the following: (1) expanded outreach by marketing to multiple networks, (2) diversity promoted among attendees, and (3) combined resources can be more cost-effective.:
Utilize multiple techniques to engage audience members at events such as word clouds, small group discussions, rotating group conversations, live polls, surveys, and narrative illustrations.
Looking Forward/Recommendations
In order to promote inclusion for people with disabilities, it is imperative that we create spaces and opportunities in the community to make sure their voices are not only heard but are the focus of the conversation. Additionally, storytelling is a relatively straightforward participatory action research (PAR) method of creating knowledge and understanding of a phenomenon. Using PAR allows the participants to shape other peoples’ minds with their own experiences, rather than the researcher doing so. As one speaker shared, telling their story was “life-changing.” It is important for researchers to consider the power of storytelling from the perspective of the person with a disability. After periods of feeling excluded, being not only included but also respected and heard can help ameliorate previous pain and can foster a sense of community. Furthermore, attending a storytelling event permits attendees to reflect and think about ways in which they or their community can be more inclusive for people with disabilities. Consider the following suggestions as you move forward in your own practice:
Strategies for Replicating in Communities
The storytelling method used in Dare2Dialogue is something that can easily be customized and implemented in other communities. Below are some strategies:
- Pick a community space that is accessible to all
- Use preferred language of the individual with the lived experience (e.g., first-person, identity-first etc.)
- Focus on a topic or interest that will engage community members with and without disabilities
- Promote community events in physical and virtual space frequeneted by people with and without disabilities
- Use community connectors to help market event
- Provide an option for attendees to reach out event coordinators for any accessibility needs (e.g., wheechair access in front of building, assistive technology, accessible parking and sign language interpreter)
- Apply recommendations of the DisABILITY Resources Tookbox (DART) for Practitioners for events (e.g., ensure access to building, provide accessible parking, use accessible modes of communication and create a welcoming environment)
Conclusion
We must create and hold space for people living with disabilities whenever we can. We read somewhere that “diversity is inviting people differently in every facet (e.g. ethnically, age, religion or spirituality, physical or mentally diverse, etc.) to the party, but “inclusion” is when they show up, you ask them to dance. We understand that some people living with a disability cannot physically stand up to dance or cognitively grasp what this means, but in this context, we mean “dance” as a metaphor for “engage in “building a relationship or community .”
DisABILITY Resources Toolbox (DART) for Practitioners
From Theory to Practice Reflections and Questions
- What does the inclusion of individuals living with a disability mean to you?
- What feelings surface for you when you think about becoming involved in community building alongside people living with any type of disability?
- In order to promote inclusion for people with disabilities, it is imperative that we create spaces and opportunities in the community to make sure their voices are not only heard but are the focus of the conversation. How would you create spaces and opportunities in the community to make sure their voices are not only heard but are the focus of the conversation?
References
Ten facts on disability. (2017). https://www.who.int/features/factfiles/disability/en/
ASDC for the Community Foundations/Intergroup Relations Program (1999). Principles for Intergroup Projects: A First Look.
Bailey, P.H. & Tilley, S. (2002). Storytelling and the interpretation of meaning in qualitative research. Journal of Advanced Nursing, 38 , 574-583. https://doi.org/10.1046/j.1365-2648.2000.02224.x
Bialik, K. (2020). 7 facts about Americans with disabilities.
Carter, E. W., & Bumble, J. L. (2018). The promise and possibilities of community conversations: Expanding opportunities for people with intellectual and developmental disabilities. Journal of Disability Policy Studies, 28 , 195–202. https://doi.org/10.1177/1044207317739408
Disabled World. (2020). Invisible Disabilities: List and General Information .
Dunn, D. S., & Andrews, E. E. (2015). Person-first and identity-first language: Developing psychologists’ cultural competence using disability language. American Psychologist, 70 (3), 255–264. https://doi.org/10.1037/a0038636
Noy, C. (2008). Sampling knowledge: The hermeneutics of snowball sampling in qualitative research. International Journal of Social Research Methodology , 11 , 327–344. https://doi.org/10.1080/13645570701401305
Okoro, C.A., Hollis, N.D., Cyrus, A.C., & Griffin-Blake, S. (2018). Prevalence of disabilities and health care access by disability status and type among adults-United States, 2016. National Library of Medicine. https://pubmed.ncbi.nlm.nih.gov/30114005/
Prilleltensky, I., & Nelson, G. (1997). Community psychology: Reclaiming social justice. In D. Fox & I. Prilleltensky (Eds.), Critical psychology: An introduction. (pp. 166– 184). Sage Publications, Inc.


6.5: Analysis of variable and absorption costing
Sales were 15,000 units in each of the three variable costing and three absorption costing income statements just presented. It was the number of units produced that varied among the three pairs of statements.
The three variable costing income statements at the different levels of production were exactly the same, each yielding operating income of $100,000, as shown in the following comparative statements.
|
15,000 units |
20,000 units |
10,000 units |
||||
|
Sales |
$750,000 |
$750,000 |
$750,000 |
|||
|
Variable cost of goods sold |
375,000 |
375,000 |
375,000 |
|||
|
Manufacturing margin |
$375,000 |
$375,000 |
$375,000 |
|||
|
Variable selling and administrative expenses |
75,000 |
75,000 |
75,000 |
|||
|
Contribution margin |
$300,000 |
$300,000 |
$300,000 |
|||
|
Fixed costs: |
||||||
|
\(\ \quad \quad\)Fixed manufacturing costs |
$150,000 |
$150,000 |
$150,000 |
|||
|
\(\ \quad \quad\)Fixed selling and administrative expenses |
50,000 |
50,000 |
50,000 |
|||
|
\(\ \quad \quad\quad \quad\)Total fixed costs |
200,000 |
200,000 |
200,000 |
|||
|
Operating income |
$100,000 |
$100,000 |
$100,000 |
The number of units manufactured during the period – 15,000; 20,000; and 10,000; respectively — does not affect operating income under the variable costing approach. This is as it should be, since production affects inventory, which is a balance sheet rather than an income statement account. When more units are produced (20,000) than sold (15,000), ending inventory is 5,000 units higher than beginning inventory. When fewer units are produced (10,000) than sold (15,000), ending inventory is 5,000 units lower than beginning inventory. Yet regardless of changes in inventory, operating income remains constant for a given level of sales because variable cost of goods sold and fixed manufacturing costs are identical for all three variable costing scenarios.
Under absorption costing, however, operating income changes when the company’s inventory balance changes. The results from the three absorption income statements presented earlier are shown again, as follows.
|
15,000 units |
20,000 units |
10,000 units |
|
|
Sales |
$750,000 |
$750,000 |
$750,000 |
|
Cost of goods sold |
575,000 |
487,500 |
575,000 |
|
Gross profit |
$175,000 |
$262,500 |
$175,000 |
|
Selling and administrative expenses |
125,000 |
125,000 |
125,000 |
|
Operating income |
$50,000 |
$137,500 |
$50,000 |
When all units manufactured (15,000) are sold (15,000), operating income under absorption costing is the same as it is under variable costing, $100,000. Under both costing methods, $150,000 of fixed factory overhead costs is deducted to arrive at operating income. It just appears in two different line items. Under variable costing, the flat amount of $150,000 follows the contribution margin line. Under absorption costing, the $150,000 is included in cost of goods sold. The fixed cost per unit is $10, determined by dividing the $150,000 total fixed factory overhead cost by the number of units produced, 15,000. The $10 per unit is then multiplied by 15,000, the number of units sold.
|
Variable costing fixed manufacturing costs |
$150,000 fixed factory overhead |
|
Absorption costing fixed manufacturing costs |
$10 fixed cost per unit x 15,000 units sold = $150,000 |
When more units are manufactured (20,000) than sold (15,000), operating income is higher under absorption costing ($137,500). Under variable costing, fixed factory overhead is the flat amount of $150,000 that follows the contribution margin line. Under absorption costing, $112,500 of fixed factory overhead cost is included in cost of goods sold. The fixed cost per unit is $7.50, determined by dividing the $150,000 total fixed factory overhead cost by the number of units produced, 20,000. The $7.50 per unit is then multiplied by 15,000, the number of units sold to get $112,500.
|
Variable costing fixed manufacturing costs |
$150,000 fixed factory overhead |
|
Absorption costing fixed manufacturing costs |
$7.50 fixed cost per unit x 15,000 units sold = $112,500 |
Since there is $37,500 less in cost of goods sold under absorption costing, there is $37,500 more operating income as a result for the same level of sales.
Conversely, when fewer units are manufactured (10,000) than sold (15,000), operating income is lower under absorption costing ($50,000). Under variable costing, fixed factory overhead is the flat amount of $150,000 that follows the contribution margin line. Under absorption costing, $225,000 of fixed factory overhead cost is included in cost of goods sold. The fixed cost per unit is $15, determined by dividing the $150,000 total fixed factory overhead cost by the number of units produced, 10,000. The $15 per unit is then multiplied by 15,000, the number of units sold to get $225,000.
|
Variable costing fixed manufacturing costs |
$150,000 fixed factory overhead |
|
Absorption costing fixed manufacturing costs |
$15 fixed cost per unit x 15,000 units sold = $225,000 |
Since there is $75,000 more in cost of goods sold under absorption costing, there is $75,000 less operating income as a result for the same level of sales.
The point of this analysis is to illustrate that under absorption costing, operating income changes based on increases or decreases in inventory due to producing more or fewer units than were sold in a period. Such changes are unrelated to a company’s operating performance, and managers need to be aware of this type of distortion under absorption costing. On a variable costing income statement, changes in inventory have no effect on operating income, making this method more reliable and desirable for analyzing profitability for an accounting period.


44.3: Quartiles and Interquartile Range
Lesson
Let's look at other measures for describing distributions.
Exercise \(\PageIndex{1}\): Notice and Wonder: Two Parties
Here are dot plots that show the ages of people at two different parties. The mean of each distribution is marked with a triangle.
What do you notice and what do you wonder about the distributions in the two dot plots?
Exercise \(\PageIndex{2}\): The Five-Number Summary
Here are the ages of the people at one party, listed from least to greatest.
\(7\qquad 8\qquad 9\qquad 10\qquad 10\qquad 11\qquad 12\qquad 15\qquad 16\qquad 20\qquad 20\qquad 22\qquad 23\qquad 24\qquad 28\qquad 30\qquad 33\qquad 35\qquad 38\qquad 42\)
-
- Find the median of the data set and label it “50th percentile.” This splits the data into an upper half and a lower half.
- Find the middle value of the lower half of the data, without including the median. Label this value “25th percentile.”
- Find the middle value of the upper half of the data, without including the median. Label this value “75th percentile.”
-
You have split the data set into four pieces. Each of the three values that split the data is called a
quartile
.
- We call the 25th percentile the first quartile . Write “Q1” next to that number.
- The median can be called the second quartile . Write “Q2” next to that number.
- We call the 75th percentile the third quartile . Write “Q3” next to that number.
- Label the lowest value in the set “minimum” and the greatest value “maximum.”
-
The values you have identified make up the
five-number summary
for the data set. Record them here.
minimum: _____ Q1: _____ Q2: _____ Q3: _____ maximum: _____ -
The median of this data set is 20. This tells us that half of the people at the party were 20 years old or younger, and the other half were 20 or older. What do each of these other values tell us about the ages of the people at the party?
- the third quartile
- the minimum
- the maximum
Are you ready for more?
There was another party where 21 people attended. Here is the five-number summary of their ages.
minimum: 5 Q1: 6 Q2: 27 Q3: 32 maximum: 60
- Do you think this party had more children or fewer children than the earlier one? Explain your reasoning.
- Were there more children or adults at this party? Explain your reasoning.
Exercise \(\PageIndex{3}\): Range and Interquartile Range
- Here is a dot plot that shows the lengths of Elena’s bus rides to school, over 12 days.
Write the five-number summary for this data set. Show your reasoning.
- The range is one way to describe the spread of values in a data set. It is the difference between the maximum and minimum. What is the range of Elena’s travel times?
- Another way to describe the spread of values in a data set is the interquartile range (IQR) . It is the difference between the upper quartile and the lower quartile.
- What is the interquartile range (IQR) of Elena’s travel times?
- What fraction of the data values are between the lower and upper quartiles?
- Here are two more dot plots.
Without doing any calculations, predict:
- Which data set has the smaller range?
- Which data set has the smaller IQR?
- Check your predictions by calculating the range and IQR for the data in each dot plot.
Summary
Earlier we learned that the mean is a measure of the center of a distribution and the MAD is a measure of the variability (or spread) that goes with the mean. There is also a measure of spread that goes with the median. It is called the interquartile range (IQR).
Finding the IQR involves splitting a data set into fourths. Each of the three values that splits the data into fourths is called a quartile .
- The median, or second quartile (Q2), splits the data into two halves.
- The first quartile (Q1) is the middle value of the lower half of the data.
- The third quartile (Q3) is the middle value of the upper half of the data.
For example, here is a data set with 11 values.
| \(12\) | \(19\) | \(20\) | \(21\) | \(22\) | \(33\) | \(34\) | \(35\) | \(40\) | \(40\) | \(49\) |
| Q1 | Q2 | Q3 |
- The median is 33.
- The first quartile is 20. It is the median of the numbers less than 33.
- The third quartile 40. It is the median of the numbers greater than 33.
The difference between the maximum and minimum values of a data set is the range . The difference between Q3 and Q1 is the interquartile range (IQR) . Because the distance between Q1 and Q3 includes the middle two-fourths of the distribution, the values between those two quartiles are sometimes called the middle half of the data .
The bigger the IQR, the more spread out the middle half of the data values are. The smaller the IQR, the closer together the middle half of the data values are. This is why we can use the IQR as a measure of spread.
A five-number summary can be used to summarize a distribution. It includes the minimum, first quartile, median, third quartile, and maximum of the data set. For the previous example, the five-number summary is 12, 20, 33, 40, and 49. These numbers are marked with diamonds on the dot plot.
Different data sets can have the same five-number summary. For instance, here is another data set with the same minimum, maximum, and quartiles as the previous example.
Glossary Entries
Definition: Interquartile Range (IQR)
The interquartile range is one way to measure how spread out a data set is. We sometimes call this the IQR. To find the interquartile range we subtract the first quartile from the third quartile.
For example, the IQR of this data set is 20 because \(50-30=20\).
| \(22\) | \(29\) | \(30\) | \(31\) | \(32\) | \(43\) | \(44\) | \(45\) | \(50\) | \(50\) | \(59\) |
| Q1 | Q2 | Q3 |
Definition: Median
The median is one way to measure the center of a data set. It is the middle number when the data set is listed in order.
For the data set 7, 9, 12, 13, 14, the median is 12.
For the data set 3, 5, 6, 8, 11, 12, there are two numbers in the middle. The median is the average of these two numbers. \(6+8=14\) and \(14\div 2=7\).
Definition: Quartile
Quartiles are the numbers that divide a data set into four sections that each have the same number of values.
For example, in this data set the first quartile is 30. The second quartile is the same thing as the median, which is 43. The third quartile is 50.
| \(22\) | \(29\) | \(30\) | \(31\) | \(32\) | \(43\) | \(44\) | \(45\) | \(50\) | \(50\) | \(59\) |
| Q1 | Q2 | Q3 |
Definition: Range
The range is the distance between the smallest and largest values in a data set. For example, for the data set 3, 5, 6, 8, 11, 12, the range is 9, because \(12-3=9\).
Practice
Exercise \(\PageIndex{4}\)
Suppose that there are 20 numbers in a data set and that they are all different.
- How many of the values in this data set are between the first quartile and the third quartile?
- How many of the values in this data set are between the first quartile and the median?
Exercise \(\PageIndex{5}\)
In a word game, 1 letter is worth 1 point. This dot plot shows the scores for 20 common words.
- What is the median score?
- What is the first quartile (Q1)?
- What is the third quartile (Q3)?
- What is the interquartile range (IQR)?
Exercise \(\PageIndex{6}\)
Mai and Priya each played 10 games of bowling and recorded the scores. Mai’s median score was 120, and her IQR was 5. Priya’s median score was 118, and her IQR was 15. Whose scores probably had less variability? Explain how you know.
Exercise \(\PageIndex{7}\)
Here are five dot plots that show the amounts of time that ten sixth-grade students in five countries took to get to school. Match each dot plot with the appropriate median and IQR.
- Median: 17.5, IQR: 11
- Median: 15, IQR: 30
- Median: 8, IQR: 4
- Median: 7, IQR: 10
- Median: 12.5, IQR: 8
Exercise \(\PageIndex{8}\)
Draw and label an appropriate pair of axes and plot the points. \(A=(10,50), B=(30,25), C=(0,30), D=(20,35)\)
(From Unit 7.3.2)
Exercise \(\PageIndex{9}\)
There are 20 pennies in a jar. If 16% of the coins in the jar are pennies, how many coins are there in the jar?
(From Unit 6.2.2)


6.5: Systems of Nonlinear Equations and Inequalities - Two Variables
In this section, you will:
- Solve a system of nonlinear equations using substitution.
- Solve a system of nonlinear equations using elimination.
- Graph a nonlinear inequality.
- Graph a system of nonlinear inequalities.
Halley’s Comet ( Figure 1 ) orbits the sun about once every 75 years. Its path can be considered to be a very elongated ellipse. Other comets follow similar paths in space. These orbital paths can be studied using systems of equations. These systems, however, are different from the ones we considered in the previous section because the equations are not linear.
In this section, we will consider the intersection of a parabola and a line, a circle and a line, and a circle and an ellipse. The methods for solving systems of nonlinear equations are similar to those for linear equations.
Solving a System of Nonlinear Equations Using Substitution
A system of nonlinear equations is a system of two or more equations in two or more variables containing at least one equation that is not linear. Recall that a linear equation can take the form Any equation that cannot be written in this form in nonlinear. The substitution method we used for linear systems is the same method we will use for nonlinear systems. We solve one equation for one variable and then substitute the result into the second equation to solve for another variable, and so on. There is, however, a variation in the possible outcomes.
Intersection of a Parabola and a Line
There are three possible types of solutions for a system of nonlinear equations involving a parabola and a line.
Possible Types of Solutions for Points of Intersection of a Parabola and a Line
Figure 2 illustrates possible solution sets for a system of equations involving a parabola and a line.
- No solution. The line will never intersect the parabola.
- One solution. The line is tangent to the parabola and intersects the parabola at exactly one point.
- Two solutions. The line crosses on the inside of the parabola and intersects the parabola at two points.
How To
Given a system of equations containing a line and a parabola, find the solution.
- Solve the linear equation for one of the variables.
- Substitute the expression obtained in step one into the parabola equation.
- Solve for the remaining variable.
- Check your solutions in both equations.
Example 1
Solving a System of Nonlinear Equations Representing a Parabola and a Line
Solve the system of equations.
- Answer
-
Solve the first equation for and then substitute the resulting expression into the second equation.
Expand the equation and set it equal to zero.
Solving for gives and Next, substitute each value for into the first equation to solve for Always substitute the value into the linear equation to check for extraneous solutions.
The solutions are and which can be verified by substituting these values into both of the original equations. See Figure 3 .
Q&A
Could we have substituted values for into the second equation to solve for in Example 1 ?
Yes, but because is squared in the second equation this could give us extraneous solutions for
For
This gives us the same value as in the solution.
For
Notice that is an extraneous solution.
Try It #1
Solve the given system of equations by substitution.
Intersection of a Circle and a Line
Just as with a parabola and a line, there are three possible outcomes when solving a system of equations representing a circle and a line.
Possible Types of Solutions for the Points of Intersection of a Circle and a Line
Figure 4 illustrates possible solution sets for a system of equations involving a circle and a line.
- No solution. The line does not intersect the circle.
- One solution. The line is tangent to the circle and intersects the circle at exactly one point.
- Two solutions. The line crosses the circle and intersects it at two points.
How To
Given a system of equations containing a line and a circle, find the solution.
- Solve the linear equation for one of the variables.
- Substitute the expression obtained in step one into the equation for the circle.
- Solve for the remaining variable.
- Check your solutions in both equations.
Example 2
Finding the Intersection of a Circle and a Line by Substitution
Find the intersection of the given circle and the given line by substitution.
- Answer
-
One of the equations has already been solved for We will substitute into the equation for the circle.
Now, we factor and solve for
Substitute the two x -values into the original linear equation to solve for
The line intersects the circle at and which can be verified by substituting these values into both of the original equations. See Figure 5 .
Try It #2
Solve the system of nonlinear equations.
Solving a System of Nonlinear Equations Using Elimination
We have seen that substitution is often the preferred method when a system of equations includes a linear equation and a nonlinear equation. However, when both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, elimination is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps. As an example, we will investigate the possible types of solutions when solving a system of equations representing a circle and an ellipse.
Possible Types of Solutions for the Points of Intersection of a Circle and an Ellipse
Figure 6 illustrates possible solution sets for a system of equations involving a circle and an ellipse .
- No solution. The circle and ellipse do not intersect. One shape is inside the other or the circle and the ellipse are a distance away from the other.
- One solution. The circle and ellipse are tangent to each other, and intersect at exactly one point.
- Two solutions. The circle and the ellipse intersect at two points.
- Three solutions. The circle and the ellipse intersect at three points.
- Four solutions. The circle and the ellipse intersect at four points.
Example 3
Solving a System of Nonlinear Equations Representing a Circle and an Ellipse
Solve the system of nonlinear equations.
- Answer
-
Let’s begin by multiplying equation (1) by and adding it to equation (2).
After we add the two equations together, we solve for
Substitute into one of the equations and solve for
There are four solutions: See Figure 7 .
Try It #3
Find the solution set for the given system of nonlinear equations.
Graphing a Nonlinear Inequality
All of the equations in the systems that we have encountered so far have involved equalities, but we may also encounter systems that involve inequalities. We have already learned to graph linear inequalities by graphing the corresponding equation, and then shading the region represented by the inequality symbol. Now, we will follow similar steps to graph a nonlinear inequality so that we can learn to solve systems of nonlinear inequalities. A nonlinear inequality is an inequality containing a nonlinear expression. Graphing a nonlinear inequality is much like graphing a linear inequality.
Recall that when the inequality is greater than, or less than, the graph is drawn with a dashed line. When the inequality is greater than or equal to, or less than or equal to, the graph is drawn with a solid line. The graphs will create regions in the plane, and we will test each region for a solution. If one point in the region works, the whole region works. That is the region we shade. See Figure 8 .
How To
Given an inequality bounded by a parabola, sketch a graph.
- Graph the parabola as if it were an equation. This is the boundary for the region that is the solution set.
- If the boundary is included in the region (the operator is or ), the parabola is graphed as a solid line.
- If the boundary is not included in the region (the operator is < or >), the parabola is graphed as a dashed line.
- Test a point in one of the regions to determine whether it satisfies the inequality statement. If the statement is true, the solution set is the region including the point. If the statement is false, the solution set is the region on the other side of the boundary line.
- Shade the region representing the solution set.
Example 4
Graphing an Inequality for a Parabola
Graph the inequality
- Answer
-
First, graph the corresponding equation Since has a greater than symbol, we draw the graph with a dashed line. Then we choose points to test both inside and outside the parabola. Let’s test the points
and One point is clearly inside the parabola and the other point is clearly outside.The graph is shown in Figure 9 . We can see that the solution set consists of all points inside the parabola, but not on the graph itself.
Graphing a System of Nonlinear Inequalities
Now that we have learned to graph nonlinear inequalities, we can learn how to graph systems of nonlinear inequalities. A system of nonlinear inequalities is a system of two or more inequalities in two or more variables containing at least one inequality that is not linear. Graphing a system of nonlinear inequalities is similar to graphing a system of linear inequalities. The difference is that our graph may result in more shaded regions that represent a solution than we find in a system of linear inequalities. The solution to a nonlinear system of inequalities is the region of the graph where the shaded regions of the graph of each inequality overlap, or where the regions intersect, called the feasible region .
How To
Given a system of nonlinear inequalities, sketch a graph.
- Find the intersection points by solving the corresponding system of nonlinear equations.
- Graph the nonlinear equations.
- Find the shaded regions of each inequality.
- Identify the feasible region as the intersection of the shaded regions of each inequality or the set of points common to each inequality.
Example 5
Graphing a System of Inequalities
Graph the given system of inequalities.
- Answer
-
These two equations are clearly parabolas. We can find the points of intersection by the elimination process: Add both equations and the variable will be eliminated. Then we solve for
Substitute the x -values into one of the equations and solve for
The two points of intersection are and Notice that the equations can be rewritten as follows.
Graph each inequality. See Figure 10 . The feasible region is the region between the two equations bounded by on the top and on the bottom.
Try It #4
Graph the given system of inequalities.
Media
Access these online resources for additional instruction and practice with nonlinear equations.


14.7: East Asian Americans
Chinese Americans are numerous in San Francisco, Oakland, the East Bay, South Bay, the Central Coast of California, Sacramento, San Diego, and the San Gabriel Valley region of Los Angeles County. The San Francisco Bay Area has a greater concentration of Cantonese-speaking Chinese than any other region in the United States. The Mexican border community of Calexico, California in addition to Mexicali has large numbers of Chinese Mexican Americans, that is, Mexican Americans of Chinese ancestry. Smaller Chinese communities can also be found in San Jacinto Valley, Lake Elsinore, and Victorville.
Southern California has perhaps the largest Taiwan-born Chinese American community in the U.S., particularly in the San Gabriel Valley, such as, Walnut, Diamond Bar, Buena Park, Cerritos, West Covina, Irvine, communities in the South Bay, Los Angeles and southern Orange County. Many minority groups from China also live in California, for example, there are Tibetan and Mongolian Americans concentrated in San Francisco, Oakland, San Jose, Orange County, and the Los Angeles/Long Beach area.
Large Korean American communities exist in the Koreatown area of Los Angeles, the eastern San Gabriel Valley, the San Fernando Valley, Cerritos/Long Beach, South Bay, Los Angeles, northern Orange County and San Diego area. There is another large Korean American population in the San Francisco Bay Area, and Koreans are growing in number in the suburban Inland Empire region, in cities such as Chino Hills, Corona, Desert Hot Springs, and Loma Linda south of San Bernardino. Since 1990, the Korean American and African American populations relocated westward and northward in the Los Angeles area.
The South Bay area and Little Tokyo have a large Japanese American community. Japanese Americans, however, are also concentrated in San Francisco and across the Bay Area, San Jose, the Salinas Valley and Santa Cruz County; and smaller communities in the Sacramento, Fresno, Bakersfield, Anaheim, San Diego, San Bernardino, Santa Barbara, and Stockton areas. Despite the presence of Japanese goods stores, media outlets and restaurants in the state, most "Little Tokyos" and "Japantowns" were evacuated during the forced relocation of Japanese Americans during World War II). As a result, most Japanese Americans in urban areas do not reside in historical Japanese communities.
Case Study - Manzanar Internment Camp
Manzanar is the site of one of ten American internment camps, where more than 120,000 Japanese Americans were incarcerated during World War II from March 1942 to November 1945. Although it had over 10,000 inmates at its peak, it was one of the smaller internment camps, situated on 6,200 acres. It is located at the foot of the Sierra Nevada mountains in California's Owens Valley, between the towns of Lone Pine to the south and Independence to the north, approximately 230 miles (370 km) north of Los Angeles. Manzanar means "apple orchard" in Spanish. The Manzanar National Historic Site, which preserves and interprets the legacy of Japanese American incarceration in the United States, was identified by the United States National Park Service as the best-preserved of the ten former camp sites.
The first Japanese Americans arrived at Manzanar in March of 1942, just one month after President Franklin D. Roosevelt signed Executive Order 9066, to build the camp their families would be staying in. Manzanar was in operation as an internment camp from 1942 until 1945. Since the last of those incarcerated left in 1945, former detainees and others have worked to protect Manzanar and to establish it as a National Historic Site to ensure that the history of the site, along with the stories of those who were incarcerated there, is recorded for current and future generations. The primary focus is the Japanese American incarceration era, as specified in the legislation that created the Manzanar National Historic Site.


3.3: Counting and Compound Events
- Recognize distinct cases when counting
- Define compound events
- Understanding events using "or", "and", and "given"
- Develop and use the Addition Rule for counting the number of possible outcomes
- Develop and use a Multiplication Rule for counting the number of possible outcomes with compound events viewed sequentially
- Count the number of possible outcomes for various compound events
Counting: Distinct Cases
Recall Text Exercise \(3.2.1.\)b, where we were drawing two tiles in sequence without replacement from a bag consisting of \(3\) red, \(2\) white, and \(1\) blue tiles. We produced our sample space by constructing a tree diagram (reproduced below).
Figure \(\PageIndex{1}\): Tree diagram of tile combinations
This tree diagram looks different from most of the tree diagrams from the previous section; two branches end with three options, but the bottom branch only has two options. The situation and the count differed based on which branch we initially started. We had to consider two cases: case \(1\) of blue initially drawn and case \(2\) of red/white initially drawn. Either case \(1\) or case \(2\) must happen because an initial tile must be drawn, but note that both cases are exclusive. If case \(1\) happens, blue is the only option for the initial draw, and then there are two options for the second tile, making our count for case \(1\) is \(1\cdot2\) \(=2.\) If case \(2\) happens, there are two options for the initial draw and three options for the second draw, making our count for case \(2\) \(2\cdot3\) \(=6.\) We have \(1 \cdot 2\) \(+ 2 \cdot 3\) \(= 8 \) possible outcomes in the sample space between the two cases.
Here is a more challenging example. Suppose a committee of ten must be formed from a group of twenty hourly employees and five managers. How many different committees can be formed if at least three managers must be on the committee? We must think carefully about how the committee is formed to count correctly. The phrase "at least three managers" tells us our selected group can have three managers or , four managers or , five managers on the committee. The committee size remains at \(10\) regardless of the number of managers. We can construct a table to determine the different committee compositions at the manager and hourly employee levels.\begin{array}{|c|c|c|c|} \hline \text{Committee Total} &10&10&10 \\ \hline \text{Number of Managers} &3&4&5 \\ \hline \text{Number of Hourly Employees} &10-3=7&10-4=6&10-5=5 \\ \hline \nonumber \end{array}
Each column of the table represents a particular committee. We recognize these committee compositions as distinct cases, one of which must happen. To count the total number of committees, we count the number of committees in each case and add them together.
\[ \nonumber \text{total number of committees}=\\ \text{number of committees with three managers}+ \text{number of committees with four managers} + \text{number of committees with five managers} \]Each case is similar in the fact that both managers and hourly employees must be selected. Our multiplication rule applies nicely since we have each group of hourly workers for each group of managers. We proceed case by case.
If the number of managers is \(3,\) we must choose \(3\) of the \(5\) managers, order of selection does not matter, yielding \( \sideset{_{5}}{_{3}}C \) different possibilities for filling the manager positions. The remaining \(7\) committee members will be chosen from the \(20\) hourly employees, yielding \( \sideset{_{20}}{_{7}}C \) different possibilities (again, we use combinations since order does not matter). Our multiplication rule informs us that there are \( \sideset{_{5}}{_{3}}C\cdot \sideset{_{20}}{_{7}}C\) \(= 10 \cdot 77,520\) \(= 775,200\) committees possible in which three of the members are managers.
Similarly for the other two cases, there are \( \sideset{_{5}}{_{4}}C\cdot \sideset{_{20}}{_{6}}C\) \(= 5 \cdot 38,760\) \(= 193,800 \) possibilities with four managers and \( \sideset{_{5}}{_{5}}C\cdot \sideset{_{20}}{_{5}}C\) \(= 1 \cdot 15,504\) \(= 15,504 \) possibilities with five managers.\[ \begin{align*}\text{number of committees with three managers}+ &\text{number of committees with four managers} + \text{number of committees with five managers} \\&= \sideset{_{5}}{_{3}}C\cdot \sideset{_{20}}{_{7}}C + \sideset{_{5}}{_{4}}C\cdot \sideset{_{20}}{_{6}}C + \sideset{_{5}}{_{5}}C\cdot \sideset{_{20}}{_{5}}C\\&= 775,200 + 193,800 + 15,504 \\ &=984,504 \end{align*}\]There are \(984,504\) possible committees that fit this description. From this work, we can also form some probability claims. If a committee is randomly formed given these restrictions, the probability that the committee will have five managers as members will be \( \frac{15,504}{984,504}\) \(\approx 1.5748\%\) and the probability that the committee will have three managers as members will be \( \frac{775,200}{984,504}\) \(\approx 78.7402\%.\)
- A food truck makes tacos with at least one but up to five fillings from eight ingredients: beef, fish, pork, beans, cheese, lettuce, tomatoes, and guacamole. How many ways can a taco be ordered from the food truck vendor?
- Answer
-
We notice that we are given restrictions on the taco fillings. We must choose at least one filling for a taco while having at most five fillings. There are several distinct cases to consider, with only one taking place.
- One filling chosen
- Two fillings chosen
- Three fillings chosen
- Four fillings chosen
- Five fillings chosen
These are the only possible choices satisfying the requirements. We can determine the counts for each of these five cases and add those counts to find the total number of possibilities. Using our earlier reasoning that each of these five cases is a combination count (order of filling choices does not matter.)\[ \nonumber \text{# of one filling}+ \text{# of two fillings} + \text{# of three fillings} + \text{# of four fillings} + \text{# of five fillings}\] \[ \nonumber \sideset{_{8}}{_{1}}C+ \sideset{_{8}}{_{2}}C + \sideset{_{8}}{_{3}}C + \sideset{_{8}}{_{4}}C + \sideset{_{8}}{_{5}}C\] \[ \nonumber 8+28+56+70+56\] \[\nonumber 218 \]A customer has \(218\) possible tacos that could be ordered. No wonder it takes some customers so long to decide what to eat.
- A ten-person committee to investigate possible corruption in U.S. military contracts is to be formed among the \(59\) members of the House of Representatives Armed Services Committee and the \(25\) members of the Senate Committee on Armed Services. The committee must have at least five House members and at least two senators. How many committees are possible?
- Answer
-
We notice that we are given restrictions on the committee structure. We must have at least five house members while still having at least two senators on the committee of ten. We have several distinct cases to consider, with only one taking place.
- Five House members and five Senators
- Six House members and four Senators
- Seven House members and three Senators
- Eight House members and two Senators
These are the only four possible committee structures satisfying the requirements. We can determine the counts for each of these four cases and sum those counts to find the total number of possible committees. We use similar reasoning to compute: \[ \nonumber \sideset{_{59}}{_{5}}C\cdot \sideset{_{25}}{_{5}}C + \sideset{_{59}}{_{6}}C\cdot \sideset{_{25}}{_{4}}C + \sideset{_{59}}{_{7}}C\cdot \sideset{_{25}}{_{3}}C + \sideset{_{59}}{_{8}}C\cdot \sideset{_{25}}{_{2}}C\] \[ \nonumber 2.66 \times 10^{11} + 5.70 \times 10^{11} + 7.85 \times 10^{11} + 6.65 \times 10^{11}\] \[\nonumber 2.29 \times 10^{12} \]There are about \(2,290,000,000,000\) committees that could be formed: an astounding number of possibilities. We are glad we only had to determine the number of possible committees and were not required to list the options by constructing a tree diagram.
Multiple counting strategies come into play as we explore possible outcomes. Counting can be pretty complicated. We are on the verge of a formal addition rule that will come in handy; we will formulate it within the context of compound events.
Connecting Events Using "or", "and", and "given"
When describing multiple events, we typically use one of the following: \(A\) or \(B,\) or \(A\) and \(B.\) Using multiple events to describe or form another event is called a compound event .
Consider rolling a single fair die. The sample space is the set: \(\{1,\) \(2,\) \(3,\) \(4,\) \(5,\) \(6\}.\)
The compound event \(\small{\text{ODD}}\) involves three outcomes: \(1,\) \(3,\) and \(5.\) We understand the event \(\small{\text{ODD}}\) to occur if when we roll the die, either a \(1,\) \(3,\) or \(5\) lands face up; there are three ways for our compound event to occur. Since the die is fair, we can compute the probability using the classical method: \(P\left(\small{\text{ODD}}\right)\) \(=\frac{3}{6}\) \(=\frac{1}{2}.\) The complement of \(\small{\text{ODD}}\) consists of the outcomes \(2,\) \(4,\) and \(6\) which we note is the compound event \(\small{\text{EVEN}};\) so, \(\overline{\small{\text{ODD}}}\) \(=\small{\text{EVEN}}.\)
The compound event \(\small{\text{PERFECT }}\)\(\small{\text{SQUARE}}\) involves just two outcomes: \(1\) and \(4.\) \(P(\small{\text{PERFECT }}\)\(\small{\text{SQUARE}})\) \(=\frac{2}{6}\) \(=\frac{1}{3}.\)
Consider the compound event \(A\) \(= \small{\text{EVEN}}\)\(\normalsize{\text{ or }}\)\( \small{\text{PERFECT }}\)\(\small{\text{SQUARE}}.\) We refer to a single compound event \(A\) using two events connected with the word "or." We understand event \(A\) to occur if, when we roll the die, either \(\small{\text{EVEN}}\) occurs or \(\small{\text{PERFECT SQUARE}}\) occurs. Meaning, if \(1,\) \(2,\) \(4,\) or \(6\) land face up, \(A\) occurs. \(P\left(A\right)\) \(=P(\small{\text{EVEN}}\)\(\normalsize{\text{ or }}\)\( \small{\text{PERFECT }}\)\(\small{\text{SQUARE}})\) \(=\frac{4}{6}\) \(=\frac{2}{3}.\)
Side note: there are different ways to consider "or". Since we include the possibility that both \(A\) and \(B\) occur, our use of the word "or" is called inclusive . If we excluded the possibility that both events occur simultaneously, the word "or" would be called exclusive . Whenever we use the word "or" in this text, we are using it inclusively.
In our last example, notice that the outcome \(4\) was described by both \(\small{\text{EVEN}}\) and \(\small{\text{PERFECT }}\)\(\small{\text{SQUARE}}.\) When two events have outcomes in common, the events are not mutually exclusive.
Consider the compound event \(B\) \(= \small{\text{EVEN}}\)\(\normalsize{\text{ and }}\)\( \small{\text{PERFECT }}\)\(\small{\text{SQUARE}}.\) We are again referring to a single compound event \(B\) using two events; this time, the two events are connected with the word "and" . We understand event \(B\) to occur if, when we roll the die, both \(\small{\text{EVEN}}\) and \(\small{\text{PERFECT SQUARE}}\) occur; meaning \(B\) will occur only when \(4\) lands face up. So, \(P\left(B\right)\) \(=P(\small{\text{EVEN}}\)\(\normalsize{\text{ and }}\)\( \small{\text{PERFECT }}\)\(\small{\text{SQUARE}})\) \(=\frac{1}{6}.\)
For a final example, consider the event \(\small{\text{EVEN}}\)\(\normalsize{\text{ given }}\)\( \small{\text{PERFECT }}\)\(\small{\text{SQUARE}}\) which is often denoted \(\small{\text{EVEN}}|\small{\text{PERFECT }}\)\(\small{\text{SQUARE}}.\) For a third time, we refer to a single event, say event \(C,\) using two events connected with the word "given." When we use "given," we treat our situation as if we had additional knowledge about what is happening; that is, we assume that the die roll is guaranteed, having a probability of \(1,\) to be a perfect square and are considering the likelihood, given this assumption, that it is even. We treat our situation conditionally by restricting our sample space to the event that follows the word "given." When examining the conditional event \(\small{\text{EVEN}}\)\(\normalsize{\text{ given }}\)\( \small{\text{PERFECT }}\)\(\small{\text{SQUARE}}\), we do not consider the original sample space of \(\{1,\) \(2,\) \(3,\) \(4,\) \(5,\) \(6\}.\) Instead, we only consider the perfect squares from that conditional sample space: \(1\) and \(4.\) We have a "new" conditional sample space of size \(2.\) We then ask, how many of these \(2\) possibilities satisfy the event \(\small{\text{EVEN}}\)? There is only one possibility of an even, \(4.\) Now in computing the probability, we must remember that we only considered the outcomes in the event \(\small{\text{PERFECT }}\)\(\small{\text{SQUARE}};\) so the total number of outcomes is not \(6\) but \(2.\) \(P(\small{\text{EVEN}}|\small{\text{PERFECT }}\)\(\small{\text{SQUARE}})\) \(=\frac{1}{2}.\)
Consider rolling two fair dice in sequence and the events below. For each event, identify the possible outcomes described and compute the probability.
Figure \(\PageIndex{2}\): Sample space of rolling two standard dice
- Event: \(\small{\text{SUM }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{TWO }}\)\(\small{\text{VALUES }}\)\(\small{\text{IS }}\normalsize7\)
- Answer
-
The sum of the values will be \(7\) for the following pairs: \(1\) and \(6,\) \(2\) and \(5,\) \(3\) and \(4,\) \(6\) and \(1,\) \(5\) and \(2,\) and \(4\) and \(3.\) We have \(P(\small{\text{SUM }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{TWO }}\)\(\small{\text{VALUES }}\)\(\small{\text{IS }}\normalsize7)\) \(=\frac{6}{36}\) \(=\frac{1}{6}.\) Notice all these outcomes fall on the diagonal from bottom left to top right.
- Event: \(\small{\text{EITHER }}\)\(\small{\text{DIE }}\)\(\small{\text{IS }}\normalsize2\)
- Answer
-
We can understand the compound event \(\small{\text{EITHER }}\)\(\small{\text{DIE }}\)\(\small{\text{IS }}\normalsize2\) as \(\small{\text{FIRST }}\)\(\small{\text{DIE }}\)\(\small{\text{IS }}\normalsize2\)\(\normalsize{\text{ or }}\)\(\small{\text{SECOND }}\)\(\small{\text{DIE }}\)\(\small{\text{IS }}\normalsize2.\) \(\small{\text{FIRST }}\)\(\small{\text{DIE }}\)\(\small{\text{IS }}\normalsize2\) consists of the \(6\) outcomes of the second row, and \(\small{\text{SECOND }}\)\(\small{\text{DIE }}\)\(\small{\text{IS }}\normalsize2\) consists of the \(6\) outcomes of second column. Note that these compound events are not mutually exclusive because the outcome where both dice are \(2\) is in both compound events. If we count the outcomes between both compound events, we arrive at \(11\) outcomes in the compound event \(\small{\text{EITHER }}\)\(\small{\text{DIE }}\)\(\small{\text{IS }}\normalsize2.\) Thus \(P(\small{\text{EITHER DIE }}\)\(\small{\text{IS }}\normalsize2)\) \(=\frac{11}{36}.\)
- Event: \(\small{\text{SUM }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{TWO }}\)\(\small{\text{VALUES }}\)\(\small{\text{IS }}\normalsize7\)\(\normalsize{\text{ and }}\)\(\small{\text{EITHER }}\)\(\small{\text{DIE }}\)\(\small{\text{IS }}\normalsize2\)
- Answer
-
We need both \(\small{\text{SUM }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{TWO }}\)\(\small{\text{VALUES }}\)\(\small{\text{IS }}\normalsize7\) and \(\small{\text{EITHER }}\)\(\small{\text{DIE }}\)\(\small{\text{IS }}\normalsize2\) to occur for our event to occur. We thus look for the overlap between the outcomes in each compound event. Only two of the outcomes from \(\small{\text{SUM }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{TWO }}\)\(\small{\text{VALUES }}\)\(\small{\text{IS }}\normalsize7\) have a \(2\) in them. Thus there are only two outcomes in the compound event \(\small{\text{SUM }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{TWO }}\)\(\small{\text{VALUES }}\)\(\small{\text{IS }}\normalsize7\)\(\normalsize{\text{ and }}\)\(\small{\text{EITHER }}\)\(\small{\text{DIE }}\)\(\small{\text{IS }}\normalsize2.\) Thus \(P(\small{\text{SUM }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{TWO }}\)\(\small{\text{VALUES }}\)\(\small{\text{IS }}\normalsize7\)\(\normalsize{\text{ and }}\)\(\small{\text{EITHER }}\)\(\small{\text{DIE }}\)\(\small{\text{IS }}\normalsize2)\) \(=\frac{2}{36}\) \(=\frac{1}{18}.\)
- Event: \(\small{\text{SUM }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{TWO }}\)\(\small{\text{VALUES }}\)\(\small{\text{IS }}\normalsize7|\small{\text{EITHER }}\)\(\small{\text{DIE }}\)\(\small{\text{IS }}\normalsize2\)
- Answer
-
We first recall that \(|\) denotes "given." Thus our event of interest is \(\small{\text{SUM }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{TWO }}\)\(\small{\text{VALUES }}\)\(\small{\text{IS }}\normalsize7\)\(\normalsize{\text{ given }}\)\(\small{\text{EITHER }}\)\(\small{\text{DIE }}\)\(\small{\text{IS }}\normalsize2.\) We know that there are only \(11\) outcomes in the event \(\small{\text{EITHER }}\)\(\small{\text{DIE }}\)\(\small{\text{IS }}\normalsize2\) and that only \(2\) of those outcomes (\(2\) and \(5,\) and \(5\) and \(2\)) add up to \(7.\) Thus \(P(\small{\text{SUM }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{TWO }}\)\(\small{\text{VALUES }}\)\(\small{\text{IS }}\normalsize7|\small{\text{EITHER }}\)\(\small{\text{DIE }}\)\(\small{\text{IS }}\normalsize2)\) \(=\frac{2}{11}.\)
- Event: \(\small{\text{EITHER }}\)\(\small{\text{DIE }}\)\(\small{\text{IS }}\normalsize2|\small{\text{SUM }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{TWO }}\)\(\small{\text{VALUES }}\)\(\small{\text{IS }}\normalsize7\)
- Answer
-
Our event of interest is \(\small{\text{EITHER }}\)\(\small{\text{DIE }}\)\(\small{\text{IS }}\normalsize2\)\(\normalsize{\text{ given }}\)\(\small{\text{SUM }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{TWO }}\)\(\small{\text{VALUES }}\)\(\small{\text{IS }}\normalsize7.\) From our previous exercise, we know that there are only \(6\) outcomes in the event \(\small{\text{SUM }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{TWO }}\)\(\small{\text{VALUES }}\)\(\small{\text{IS }}\normalsize7\) and that only \(2\) of those outcomes, \(2\) and \(5,\) and \(5\) and \(2,\) include \(2.\) Thus \(P(\small{\text{EITHER }}\)\(\small{\text{DIE }}\)\(\small{\text{IS }}\normalsize2|\small{\text{SUM }}\)\(\small{\text{OF }}\)\(\small{\text{THE }}\)\(\small{\text{TWO }}\)\(\small{\text{VALUES }}\)\(\small{\text{IS }}\normalsize7)\) \(=\frac{2}{6}\) \(=\frac{1}{3}.\)
Some statisticians and educators introduce the term simple event to help students better handle complex event descriptions. The general idea is that we can better understand or more easily compute probabilities when complex event descriptions are understood using combinations of other most basic events called simple events. The difficulty in such a presentation lies in the subjective and experiential aspects of understanding descriptions as basic and straightforward. To avoid the technical difficulty of a sufficient definition and application, we avoid the term throughout the book but, hopefully, still manage to instill the underlying concept.
Compound Events and Counting
Again, we build our intuition by counting when rolling a single fair die. When we were considering the compound event \(\small\text{EVEN}\)\(\normalsize\text{ or }\)\(\small\text{PERFECT }\)\(\small\text{SQUARE},\) computing the probability boiled down to two counting questions: how large is the sample space and how many outcomes comprise the compound event? The first counting question has an answer of \(6.\) Having already worked the problem, we know the answer to the second question is \(4,\) but how does this relate to the two separate events that form our compound event? In counting the outcomes in \(\small\text{EVEN},\) we arrive at \(3,\) and the number of outcomes in \(\small\text{PERFECT }\)\(\small\text{SQUARE}\) is \(2.\) It would appear that the number of outcomes in \(\small\text{EVEN}\)\(\normalsize\text{ or }\)\(\small\text{PERFECT }\)\(\small\text{SQUARE}\) is \(3+2\) \(=5.\) Our previous work shows that the answer is \(4.\) We must have counted some outcome(s) twice, which can happen whenever the events are not mutually exclusive. Recall that the outcome \(4\) is both even and a perfect square but should only be counted once. The outcomes that are counted twice are in both \(\small\text{EVEN}\) and \(\small\text{PERFECT }\)\(\small\text{SQUARE}.\) The total count is the number of outcomes in \(\small\text{EVEN}\) plus the number of outcomes in \(\small\text{PERFECT }\)\(\small\text{SQUARE}\) minus the number of outcomes in \(\small\text{EVEN}\)\(\normalsize\text{ and }\)\(\small\text{PERFECT }\)\(\small\text{SQUARE}\); namely, \(3+2-1\) \(=4.\)
Formally, this is known as the Addition Rule for Counting . Given events \(A\) and \(B,\)\[\nonumber \text{# of outcomes in }(A \text{ or }B)=\text{# of outcomes in }A+\text{# of outcomes in }B-\text{# of outcomes in }(A \text{ and }B)\]
Explain how we used the Addition Rule for Counting at the beginning of this section when counting tiles and committees.
- Answer
-
We were attempting to determine the many outcomes in a sample space. Each case can be identified as a compound event, and we joined these compound events using the word "or." We added each of the counts together to get the total sum. This looks similar to the addition rule, except we never had to subtract anything since the cases were mutually exclusive; no two cases shared an outcome. This would be like subtracting \(0\) outcomes. The addition rule was at play the whole time.
Greater care must be taken when counting outcomes involving compound events formed using "and." We cannot mindlessly look to some formula that will always work; we must assess the situation and check that certain conditions are met. To develop our intuition regarding counting \(A\text{ and }B,\) we turn our attention to a more robust but familiar example, rolling two fair dice in sequence, and analyze: \(\small\text{EVEN }\)\(\small\text{FIRST}\)\(\normalsize{\text{ and }}\)\(\small\text{PERFECT }\)\(\small\text{SQUARE }\)\(\small\text{SECOND}.\)
Context matters tremendously; read carefully and understand the current situation. We have dealt with the events \(\small\text{EVEN}\) and \(\small\text{PERFECT }\)\(\small\text{SQUARE}\) in the context of a roll of a single fair die. The sample space consists of \(36\) pairs of dice values; we are counting the number of outcomes with an even value for the first die and a perfect square for the second die. The multiplication rule worked well for determining the size of sample space because we could understand our outcomes as pairs of outcomes in sequence. Suppose we can understand \(\small\text{EVEN }\)\(\small\text{FIRST}\)\(\normalsize{\text{ and }}\)\(\small\text{PERFECT }\)\(\small\text{SQUARE }\)\(\small\text{SECOND}\) as two events in sequence, the same ideas apply.  We arrive at the conclusion there are \(3\cdot2\) \(=6\) outcomes in \(\small\text{EVEN }\)\(\small\text{FIRST}\)\(\normalsize{\text{ and }}\)\(\small\text{PERFECT }\)\(\small\text{SQUARE }\)\(\small\text{SECOND}\). For this example, we can easily construct the outcomes to confirm our count: \(2\) and \(1,\) \(4\) and \(1,\) \(6\) and \(1,\) \(2\) and \(4,\) \(4\) and \(4,\) and \(6\) and \(4.\)
Within the context of rolling two die in sequence, determine the number of outcomes in \(\small\text{ONE }\)\(\small\text{DIE }\)\(\small\text{IS }\)\(\small\text{EVEN }\)\(\small\text{WHILE }\)\(\small\text{THE }\)\(\small\text{OTHER }\)\(\small\text{IS }\)\(\small\text{A }\)\(\small\text{PERFECT }\)\(\small\text{SQUARE}.\)
- Answer
-
To save space, let \(\small{\text{EVEN}}\)\(=E\) and \(\small\text{PERFECT }\)\(\small\text{SQUARE}\)\(=PS.\) We can understand the event as follows: \((E\small{\text{ FIRST}}\)\(\normalsize\text{ and }\)\(PS \small{\text{ SECOND}}\)\(\normalsize) \text{ or }\)\( (PS\small{\text{ FIRST}}\)\(\normalsize\text{ and }\)\(E \small{\text{ SECOND}}\normalsize)\). We can count the number of outcomes using both addition and multiplication rules. Applying the same logic as in the previous example and recognizing that \(4\) satisfies both events, yields:\[\text{# in }E\text{ and }PS\\=\text{# in } (E\small{\text{ FIRST }}\normalsize\text{ and }PS\small{\text{ SECOND }}\normalsize)+\text{# in } (PS\small{\text{ FIRST }}\normalsize\text{ and }E \small{\text{ SECOND }}\normalsize)-\text{# in }((E\small{\text{ FIRST }}\normalsize\text{ and }PS \small{\text{ SECOND }}\normalsize)\text{ and }(PS\small{\text{ FIRST }}\normalsize\text{ and }E \small{\text{ SECOND }}\normalsize))\\=3\cdot2+2\cdot3-1\\=6+6-1=11\nonumber\]
Note that it might be tempting to describe \(\small\text{ONE }\)\(\small\text{DIE }\)\(\small\text{IS }\)\(\small\text{EVEN }\)\(\small\text{WHILE }\)\(\small\text{THE }\)\(\small\text{OTHER }\)\(\small\text{IS }\)\(\small\text{A }\)\(\small\text{PERFECT }\)\(\small\text{SQUARE}\) as the compound event \(\small{\text{EVEN}}\)\(\normalsize{\text{ and }}\)\(\small\text{PERFECT }\)\(\small\text{SQUARE}\). In the context of rolling two dice, we may interpret the event differently. For example, rolling a double \(4\) (satisfying both events), one die is even, and the other is a perfect square (one die satisfying one and the other die satisfying the other), or as long as both conditions are met between the two dice, the event occurs (a \(4\) with any other die value). Clarity in communication is key; put thought into the event description to protect it from ambiguity to the best of your ability.
To conclude this section and formalize one last component of counting compound events, we consider a familiar context in probability: drawing cards from a standard deck of playing cards. We draw two cards in sequence from a single deck of standard playing cards without replacing the cards drawn and examine the compound event \(\small\text{SPADE }\)\(\small\text{FIRST}\)\(\normalsize{\text{ and }}\)\(\small\text{BLACK }\)\(\small\text{CARD }\)\(\small\text{SECOND}.\)
Our counting procedure closely models the counting in the previous section. The phrasing of the compound event helps us readily understand our compound event as consisting of pairs of outcomes in sequence and uses the multiplication rule. There is, however, one stark contrast. In rolling two fair die and considering \(\small\text{EVEN }\)\(\small\text{FIRST}\)\(\normalsize{\text{ and }}\)\(\small\text{PERFECT }\)\(\small\text{SQUARE }\)\(\small\text{SECOND},\) the value of the first die did not affect the possible values of the second die. Whatever card we draw first cannot be drawn again in our current context. There is a dependency between the events. This does not hinder our ability to count using the multiplication rule; we have run into this idea before in Text Exercise \(3.2.2.\)b. There are \(13\) spades and \(26\) black cards in a standard deck of playing cards. There are \(13\) ways for \(\small\text{SPADE }\)\(\small\text{FIRST}\) to happen. Since we are counting the number of ways \(\small\text{SPADE }\)\(\small\text{FIRST}\)\(\normalsize{\text{ and }}\)\(\small\text{BLACK }\)\(\small\text{CARD }\)\(\small\text{SECOND}\) can happen, we count the number of ways our second card can be black given our first card was a spade. Since spades are black, only \(25\) black cards are left. We have the total number of outcomes \(13\cdot 25\) \(=325.\)
We provide a Multiplication Rule for Counting Compound Events with "and." Given events \(A\) and \(B,\)\[\nonumber \text{# of outcomes in }A\text{ and }B= (\text{# outcomes in A})\cdot(\text{# outcomes in B|A})\]
Within the context of drawing two cards in sequence from a single deck of standard playing cards without replacing the cards drawn, determine the number of outcomes in the event \(\small\text{BLACK }\)\(\small\text{CARD }\)\(\small\text{FIRST}\)\(\normalsize{\text{ and }}\)\(\small\text{SPADE }\)\(\small\text{ SECOND}.\)
- Answer
-
We are in a similar situation to the previous example; however, the order of the events has switched. We can easily understand the events as in sequence and apply the multiplication rule.\[\nonumber \text{# of outcomes in }\small{\text{BLACK CARD FIRST}}\normalsize{\text{ and }}\small{\text{SPADE SECOND}}\normalsize= (\text{# outcomes in }\small{\text{BLACK CARD FIRST}}\normalsize)\cdot(\text{# outcomes in }\small{\text{SPADE SECOND}}|\small{\text{BLACK CARD FIRST}}\normalsize)\]The count in the second term depends on whether or not a spade was drawn the first time. We, therefore, need to reformulate our approach. The event \(\small{\text{BLACK CARD}}\) consists of all the spades and clubs; so we can think of the event \(\small{\text{BLACK CARD}}\) as the compound event \(\small{\text{SPADE}}\) or \(\small{\text{CLUB}}\). Let us refer to these events as \(S\) and \(C,\) respectively. So we can understand our event as follows:\[\nonumber (\small{\text{S or C FIRST}}\normalsize{) \text{ and }}\small{\text{S SECOND}}\\(\small{\text{S FIRST}}\normalsize{\text{ and }}\small{\text{S SECOND}}\normalsize{\text{) or (}}\small{\text{C FIRST}}\normalsize{ \text{ and }}\small{\text{S SECOND}}\normalsize{)}\]We can thus apply both addition and multiplication rules to compute our total number of outcomes.\[\nonumber \text{# of outcomes in } \small{\text{S FIRST}}\normalsize{\text{ and }}\small{\text{S SECOND}}=13\cdot12=156 \\ \text{# of outcomes in }\small{\text{C FIRST}}\normalsize{ \text{ and }}\small{\text{S SECOND}}=13\cdot13=169\\ \text{# of outcomes in } (\small{\text{S FIRST}}\normalsize{\text{ and }}\small{\text{S SECOND}}\normalsize{\text{) and (}}\small{\text{C FIRST}}\normalsize{ \text{ and }}\small{\text{S SECOND}}\normalsize{)}=0\]Thus the total number of outcomes is \(156+169-0=\) \(325.\)
A family of \(5\) is attending a convention on family life. The theme of this year's convention is nature and quality time. The opening banquet will have \(4\) door prizes related to the current theme.  Any person in attendance can win at most one prize. The family of \(5\) recently invested in their backyard patio with new furniture and have a trampoline but are very interested in the other two prizes. If there will be \(400\) people in attendance, how many ways can the door prizes be awarded so that this family gets the first and the fourth prizes?
- Answer
-
While winning quality prizes even if you do not need them can be exciting, the question excludes the case where the family wins all four prizes. The event of interest is \(\small\text{WINNING }\)\(\small\text{FIRST }\)\(\text{and }\)\(\small\text{LOSING }\)\(\small\text{SECOND }\)\(\text{and }\)\(\small\text{LOSING }\)\(\small\text{THIRD }\)\(\text{and }\)\(\small\text{WINNING }\)\(\small\text{FOURTH.}\) There are \(5\) ways of winning the first prize (one of the five family members must be chosen the winner), \(395\) ways of not winning the second prize given a successful win of the first prize, and \(394\) ways of not winning the third prize given the desired outcomes of the first and second prizes, and just \(4\) ways to win the last prize given the outcomes of the first three prizes. There are \(5\cdot395\cdot394\cdot4\) \(=3,112,600\) possibilities. This could happen in many ways, but what is the probability that it does happen? There are \(400\cdot399\cdot398\cdot397\) \(=25,217,757,600\) ways the prizes could be awarded. The probability of this becoming a reality is very small. \(\frac{3,112,600}{25,217,575,600}\) \(\approx0.01234\%.\) It would be highly unlikely for the family to win the first and fourth prizes.


9.13: Current Issues in Corrections- War on Drugs and Gangs
Transcarceration
Get Tough Policies
[1] This policy increased sentence lengths for 40 felonies, which included life imprisonment. Perhaps the largest 3 strikes policy was in 1994, in California, with Proposition 184, commonly called the Three Strikes and You’re Out policy. It mandated a minimum of 25 years of prison for individuals committing 3 felonies. What made this policy more pervasive than others was the way in which it could be applied. If a person had two previous strikes for violent, or serious felonies (not necessarily violent), any new felony was life imprisonment, with a minimum of 25 years. For a more detailed view of this policy, see https://lao.ca.gov/2005/3_strikes/3_strikes_102005.htm
Federal Drug Inmates
-
Wright, P. (1995). Three strikes racks ‘em up.
Journal of Prisoners on Prisons, 6
(2), 3-6.
https://www.ncjrs.gov/App/Publications/abstract.aspx?ID=162918
↵
-
Meierhoefer, B. (1992). The general effect of mandatory minimum prison terms: A longitudinal study of federal sentences imposed.
Federal Judicial Center
. www.fjc.gov/sites/default/files/2012/GenEffMM.pdf
↵


From
Learning with ‘e’s: My thoughts about learning technology and all things digital. (March 28, 2020)
Ever wondered how we can overcome feelings of loneliness and disconnect in distance education? It’s a question that online educators have been grappling with for a long time.
When you’re online, or using your smartphone to communicate, do you ever feel ‘connected’ to the person at the other end? It’s a common psychological phenomenon to feel that although you are separated by distance, the technology actually mediates your connection with someone else. Conversely, if you feel disconnected, remote or in some way out of tune with the person at the other end, dialogue can be limited, and connection brief. This is where the technology may have failed to support the interaction.
Feelings of intimacy, or warmth, or common understanding through technology all fall under the category of
social presence
. It’s a term psychologists and technologists use to describe the ability to project physical, social and emotional presence and also to experience it from others during interactions. It’s almost like being there alongside them.
Short, Williams and Christie (1976) argued that there is a spectrum of social presence inherent in the affordances of available technologies. That is, some technologies are better at creating the conditions for good social presence than others. When I was conducting the research for my research thesis (Wheeler, 2007), I used this principle to differentiate between the pedagogical power of four modes of communication: face to face (the richest), video conference, telephone, and e-mail (the poorest).
Today we have many more communication technologies to call upon, including handheld videoconferencing (Facetime and Whatsapp for example), social media in various formats, online discussion groups, virtual reality, social gaming, and other options still emerging. Teachers who wish to use these technologies to support learning at a distance need to realise that each of these tools have various
affordances
(Gibson, 1966) some of which are more conducive than others, depending on the activities they are supporting.
So, when designing online activities and content, it is useful for educators to consider firstly what technologies are available to the student and secondly which of these technologies is best suited to support the activities. Almost always, social presence is a deciding factor in whether students persist in their remote studies, or whether they give up (Wheeler, 2007).
(Image: Young man seen from behind walking alone by Richard Revel,
https://www.publicdomainpictures.net...re=lonely-walk
, Public Domain)


3.1: Introduction
Health promotion, disease and injury prevention, and well-person care are essential components of comprehensive health care for persons assigned female at birth (AFAB) and those identifying as female. These persons experience unique and specific physical and mental health issues at various stages and events in their lives, and promoting their well-being requires a comprehensive approach that includes health education, the promotion of good nutrition and healthy self-care practices, annual checkups, preventive screenings and immunizations, contraception counseling, preconception care, perinatal care, and menopause management. By prioritizing these goals, health-care professionals and policymakers can improve overall health outcomes, reduce health disparities, and empower these persons to take control of their health. This chapter aims to provide insights and strategies for obstetric and gynecologic nurses in their critical role in promoting their patients’ health and well-being throughout the lifespan.


1.1: Introduction
There are three principal themes to our course:
Discrete Structures Graphs, digraphs, networks, designs, posets, strings, patterns, distributions, coverings, and partitions.
Enumeration Permutations, combinations, inclusion/exclusion, generating functions, recurrence relations, and Pólya counting.
Algorithms and Sorting, eulerian circuits, hamiltonian cycles, planarity testing, graph coloring, spanning trees, shortest paths, network flows,
Optimization bipartite matchings, and chain partitions.
To illustrate the accessible, concrete nature of combinatorics and to motivate topics that we will study, this preliminary chapter provides a first look at combinatorial problems, choosing examples from enumeration, graph theory, number theory, and optimization. The discussion is very informal—but this should serve to explain why we have to be more precise at later stages. We ask lots of questions, but at this stage, you'll only be able to answer a few. Later, you'll be able to answer many more … but as promised earlier, most likely you'll never be able to answer them all. And if we're wrong in making that statement, then you're certain to become very famous. Also, you'll get an A++ in the course and maybe even a Ph.D. too.