6.2: Converting a Decimal to a Fraction
- be able to convert an ordinary decimal and a complex decimal to a fraction
Converting an Ordinary Decimal to a Fraction
We can convert a decimal fraction to a fraction, essentially, by saying it in words, then writing what we say. We may have to reduce that fraction.
Convert each decimal fraction to a proper fraction or a mixed number.
Solution
Reading: six tenths \(\to \dfrac{6}{10}\).
Reduce: \(\dfrac{3}{5}\)
Solution
Reading: nine hundred three thousands \(\to \dfrac{903}{1000}\).
Solution
Reading: eighteen and sixty-one hundredths \(\to 18 \dfrac{61}{100}\).
Solution
Reading: five hundred eight and five ten thousandths \(\to 508 \dfrac{5}{10,000}\).
Reduce: \(508 \dfrac{1}{2,000}\).
Practice Set A
Convert the following decimals to fractions or mixed numbers. Be sure to reduce.
16.84
- Answer
-
\(16 \dfrac{21}{25}\)
Practice Set A
0.513
- Answer
-
\(\dfrac{513}{1,000}\)
Practice Set A
6,646.0107
- Answer
-
\(6,646 \dfrac{107}{10,000}\)
Practice Set A
1.1
- Answer
-
\(1 \dfrac{1}{10}\)
Converting A Complex Decimal to a Fraction
Numbers such as \(0.11 \dfrac{2}{3}\) are called complex decimals . We can also convert complex decimals to fractions.
Convert the following complex decimals to fractions.
\(0.11 \dfrac{2}{3}\)
Solution
The \(\dfrac{2}{3}\) appears to occur in the thousands position, but it is referring to \(\dfrac{2}{3}\) of a hundredth. So, we read \(0.11 \dfrac{2}{3}\) as "eleven and two-thirds hundredths."
\(\begin{array} {rcl} {0.11 \dfrac{2}{3} = \dfrac{11 \dfrac{2}{3}}{100}} & = & {\dfrac{\dfrac{11 \cdot 3 + 2}{3}}{100}} \\ {} & = & {\dfrac{\dfrac{35}{3}}{\dfrac{100}{1}}} \\ {} & = & {\dfrac{35}{3} \div \dfrac{100}{1}} \\ {} & = & {\dfrac{\begin{array} {c} {^7} \\ {\cancel{35}} \end{array}}{3} \cdot \dfrac{1}{\begin{array} {c} {\cancel{100}} \\ {^{20}} \end{array}}} \\ {} & = & {\dfrac{7}{60}} \end{array}\)
\(4.006 \dfrac{1}{4}\)
Solution
Note that \(4.006 \dfrac{1}{4} = 4 + .006 \dfrac{1}{4}\)
\(\begin{array} {rcl} {4 + .006 \dfrac{1}{4}} & = & {4 + \dfrac{6 \dfrac{1}{4}}{1000}} \\ {} & = & {4 + \dfrac{\dfrac{25}{4}}{\dfrac{1000}{1}}} \\ {} & = & {4 + \dfrac{\begin{array} {c} {^1} \\ {\cancel{25}} \end{array}}{4} \cdot \dfrac{1}{\begin{array} {c} {\cancel{1000}} \\ {^{40}} \end{array}}} \\ {} & = & {4 + \dfrac{1 \cdot 1}{4 \cdot 40}} \\ {} & = & {4 + \dfrac{1}{160}} \\ {} & = & {4 \dfrac{1}{160}} \end{array}\)
Practice Set B
Convert each complex decimal to a fraction or mixed number. Be sure to reduce.
\(0.8 \dfrac{3}{4}\)
- Answer
-
\(\dfrac{7}{8}\)
Practice Set B
\(0.12 \dfrac{2}{5}\)
- Answer
-
\(\dfrac{31}{250}\)
Practice Set B
\(6.005 \dfrac{5}{6}\)
- Answer
-
\(6 \dfrac{7}{1,200}\)
Practice Set B
\(18.1 \dfrac{3}{17}\)
- Answer
-
\(18 \dfrac{2}{17}\)
Exercises
For the following 20 problems, convert each decimal fraction to a proper fraction or a mixed number. Be sure to reduce.
Exercise \(\PageIndex{1}\)
0.7
- Answer
-
\(\dfrac{7}{10}\)
Exercise \(\PageIndex{2}\)
0.1
Exercise \(\PageIndex{3}\)
0.53
- Answer
-
\(\dfrac{53}{100}\)
Exercise \(\PageIndex{4}\)
0.71
Exercise \(\PageIndex{5}\)
0.219
- Answer
-
\(\dfrac{219}{1,000}\)
Exercise \(\PageIndex{6}\)
0.811
Exercise \(\PageIndex{7}\)
4.8
- Answer
-
\(4 \dfrac{4}{5}\)
Exercise \(\PageIndex{8}\)
2.6
Exercise \(\PageIndex{9}\)
16.12
- Answer
-
\(16 \dfrac{3}{25}\)
Exercise \(\PageIndex{10}\)
25.88
Exercise \(\PageIndex{11}\)
6.0005
- Answer
-
\(6 \dfrac{1}{2,000}\)
Exercise \(\PageIndex{12}\)
1.355
Exercise \(\PageIndex{13}\)
16.125
- Answer
-
\(16 \dfrac{1}{8}\)
Exercise \(\PageIndex{14}\)
0.375
Exercise \(\PageIndex{15}\)
3.04
- Answer
-
\(3 \dfrac{1}{25}\)
Exercise \(\PageIndex{16}\)
21.1875
Exercise \(\PageIndex{17}\)
8.225
- Answer
-
\(8 \dfrac{9}{40}\)
Exercise \(\PageIndex{18}\)
1.0055
Exercise \(\PageIndex{19}\)
9.99995
- Answer
-
\(9 \dfrac{19,999}{20,000}\)
Exercise \(\PageIndex{20}\)
22.110
For the following 10 problems, convert each complex decimal to a fraction.
Exercise \(\PageIndex{21}\)
\(0.7 \dfrac{1}{2}\)
- Answer
-
\(\dfrac{3}{4}\)
Exercise \(\PageIndex{22}\)
\(0.012 \dfrac{1}{2}\)
Exercise \(\PageIndex{23}\)
\(2.16 \dfrac{1}{4}\)
- Answer
-
\(2 \dfrac{13}{80}\)
Exercise \(\PageIndex{24}\)
\(5.18 \dfrac{2}{3}\)
Exercise \(\PageIndex{25}\)
\(14.112 \dfrac{1}{3}\)
- Answer
-
\(14 \dfrac{337}{3,000}\)
Exercise \(\PageIndex{26}\)
\(80.0011 \dfrac{3}{7}\)
Exercise \(\PageIndex{27}\)
\(1.40 \dfrac{5}{16}\)
- Answer
-
\(1 \dfrac{129}{320}\)
Exercise \(\PageIndex{28}\)
\(0.8 \dfrac{5}{3}\)
Exercise \(\PageIndex{29}\)
\(1.9 \dfrac{7}{5}\)
- Answer
-
\(2 \dfrac{1}{25}\)
Exercise \(\PageIndex{30}\)
\(1.7 \dfrac{37}{9}\)
Exercises for Review
Exercise \(\PageIndex{31}\)
Find the greatest common factor of 70, 182, and 154.
- Answer
-
14
Exercise \(\PageIndex{32}\)
Find the greatest common multiple of 14, 26, and 60.
Exercise \(\PageIndex{33}\)
Find the value of \(\dfrac{3}{5} \cdot \dfrac{15}{18} \div \dfrac{5}{9}\).
- Answer
-
\(\dfrac{9}{10}\)
Exercise \(\PageIndex{34}\)
Find the value of \(5 \dfrac{2}{3} + 8 \dfrac{1}{12}\).
Exercise \(\PageIndex{35}\)
In the decimal number 26.10742, the digit 7 is in what position?
- Answer
-
thousandths


4.4: Why not use the notion of needs?
By introducing the concepts of ‘functionings’ and ‘capabilities’, the capability approach offers some specific notions of ‘advantage’ and provides an ethical framework to guide our actions and institutional design. It is also a theoretical framework with clear commitments to practice and policy making in the world as it is, not just in some hypothetical world or in a stylized model. However, the same can be said of the (basic) needs approach, which was introduced and developed much earlier in the landscape of ethical approaches related to wellbeing and poverty. Hence, the obvious question to ask is: why not use the notion of needs for our theoretical work, and the basic needs approach for work on development?
To answer that question, it makes sense to make a distinction between the basic needs approach as it has been used by development scholars and policy makers, and the philosophical theories of basic needs. Let us look at the basic needs approach first. This is a practice and policy oriented approach “that gives priority to meeting people’s basic needs — to ensuring that there are sufficiently, appropriately distributed basic need goods and services to sustain all human lives at a minimally decent level” (Stewart 2006, 14). The basic needs approach was a reaction to the development policies that many countries in the Global South pursued after independence from colonial rule, and that led to a dualistic pattern of development, with a small modern sector that allowed some people to flourish while at the same time leaving many other people in poverty and unemployment. The fundamental claim of the basic needs approach was that the poor not only need a monetary income but also some very basic goods and services such as clean water, enough food, health services, and education. Given the urgency of these needs, the hope was that the requirement to supply them would be more readily accepted by governments in both the Global South and North than theoretical arguments about inequality (Stewart 2006, 15).
In the 1980s, the basic needs approach lost support because development donors shifted their attention to the goals of stability and adjustment, but when they again started to pay attention to the poor, adopting the capability approach, and especially the more policy oriented human development paradigm, seemed more attractive. However, according to Frances Stewart (2006, 18), when applied to the concern of reducing poverty in the Global South, the capability approach and the basic needs approach are very similar in terms of the actions they recommend.
Why, then, would anyone consider the capability approach over the basic needs approach? The first reason is that the capability approach seems to have a more elegant philosophical foundation (Stewart 2006, 18). However, while its seeming elegance might have been perceived by the basic needs practitioners as a reason for its adoption, the question is whether this is true, given that there have always been the philosophical theories of needs, to which we turn below. Perhaps the answer is pragmatic and points to an advantage of the interdisciplinary nature of the capability approach: Sen did not only lay out its theoretical foundations, but also, via his empirical work and his contributions to the Human Development Reports, translated those philosophical ideas into practice. Perhaps it is the case that a philosophical theory needs some charismatic thinker who translates it into practice, since otherwise it is not picked up by policy makers. In the case of the capability approach, Sen did both the philosophical work and the policy translation.
Another reason mentioned by Stewart is that the capability approach focuses more on the situation of individuals than the basic needs approach (Stewart 2006, 18). The capability approach doesn’t recommend the delivery of the same basic goods to everyone, but rather that we take human diversity as much as possible into account. The basic needs approach was more broad-brush than the capability approach, which stresses the additional resources needed by some people, such as the disabled. A third reason is that the capability approach applies to all human beings, hence also to the rich, whereas the basic needs approach has generally been perceived as focused on poor people in poor countries (Streeten 1995, ix). The more inclusive scope of the capability approach, which applies to all human beings, resonates with an increasing acknowledgement that countries in the Global North also include people with low levels of wellbeing, and that ideas of development, social progress and prosperity apply to all countries. This is very well captured in the case of the Sustainable Development Goals, which are goals applicable to all countries.
Many of the earlier key advocates of the basic need approach are pursuing their goals now under the umbrella of the human development paradigm. Hence the pragmatic and policy oriented basic needs approach has joined the human development paradigm, which has become much stronger politically. For policy making, the influence of frameworks at a particular point in time is one of the relevant considerations whether one should adopt one framework rather than another, and hence there is a good reason why, given their pragmatic goals, the basic needs advocates have contributed to a joint endeavour with capability scholars to set up the human development paradigm.
But what about the philosophical theories of basic needs? Are there reasons why we should favour them rather than the capability approach? The arguments that were given for the pragmatic basic needs approach apply also to some extent to the questions of the complementarities and differences between the theories. There are, theoretically, close similarities between theories of needs and capabilities, and Soran Reader (2006) has argued that many of the objections that capability scholars have to theories of needs are unwarranted and based on implausibly reductionist readings of theories of needs. According to Reader, theories of needs and capability theories have much more in common than capability scholars have been willing to see.
Still, Sen has been notorious in arguing that the capability approach is superior to the basic needs approach, a critique he has reaffirmed in his latest book (Sen 2017, 25). In his paper ‘Goods and people’, Sen (1984a, 513–15) criticised the basic needs approach for being too focussed on commodities, and seeing human beings as passive and needy. Those criticisms were rebutted by Alkire (2002, 166–74), who believed they were based on misinterpretations. However, Alkire did argue that the other two claims by Sen were correct. First, that the basic needs approach confines our attention to the most desperate situations, and is therefore only useful to developing countries. Alkire, however, sees this as potentially a strength of the basic needs approach; one could argue that it helps us to focus our attention on the worst off. Sen’s final criticism was, according to Alkire, the one with most theoretical bite: that the basic needs approach does not have solid philosophical foundations.
However, the question is whether that is true. A set of recent papers by basic needs scholars (Brock and Reader 2002; Reader and Brock 2004; Reader 2006) make clear that the philosophical theory of basic needs is sophisticated; moreover, several philosophically highly sophisticated theories of needs have been proposed in the past, both in the Aristotelian tradition but also more recently by contemporary philosophers (e.g. Doyal and Gough 1991; Wiggins 1998). Instead, a more plausible explanation for the basic needs approach losing ground in comparison to the capability approach seems to me that in the case of the basic needs approach, there was less interaction between empirical scholars and policy makers on the one hand, and philosophers on the other. It is hard to find evidence of a clear synergy between basic needs philosophers and basic needs development scholars and policy advisors — an interaction that is very present in the capability literature. This may explain why the basic needs approach has been seen as lacking solid conceptual foundations.
Yet despite these hypotheses, which may help us understand why the capability approach to a large extent replaced the focus on needs in the practical field and in empirical research, some genuine differences remain. The first difference requires the capability approach to adopt some basic distinctions that are fundamental to philosophical needs theory, and for which the capability approach in itself does not have the resources: the distinction between non-contingent needs on the one hand, and contingent needs, desires, wants, etc. on the other hand. Non-contingent needs are cases in which “the needing being simply cannot go on unless its need is met” (Reader and Brock 2004, 252). This relates to a more common-sense distinction between ‘needs’ and ‘wants’, that has an important relevance to our everyday ethical life and to policy making, but that has no equivalent in the capability approach. For some applications of the capability approach, such as those related to prioritising in conditions of extreme scarcity of resources (whether these resources are money, food, water, the right to emit greenhouse gasses, etc.) theories of needs can provide tools to guide our moral priorities that are lacking in the capability approach. Right now, preferences dominate in public decision making, but the concept of preferences cannot make a distinction between a preference for minimal amounts of water, food, safety, and social interaction, versus a preference for wine and a jacuzzi. The preferences-based approach, which has become very dominant in ethical theory as well as policy analysis by economists, doesn’t have the theoretical resources to make such a distinction, whereas it is central to some of our intuitions of how to prioritise our actions in cases of scarcity (e.g. J. O’Neill 2011; Robeyns 2017a).
The second difference follows from the first. The distinction in theories of needs between the morally required (meeting non-contingent or basic needs) and the morally laudable but not required (the other needs, wants, desires, etc.) implies that the needs approach may have a smaller scope than the capability approach. The modular view of the capability approach presented in chapter 2 makes clear that the capability approach can be used for a wide variety of capabilitarian theories and applications. The basic needs approach is more focussed on situations in which we need to prioritise — but the domain of ethical questions is broader than that.
In conclusion, the basic needs approach in practice is, for pragmatic and political reasons, now part of the human development paradigm. At the theoretical level, though, capability scholars neglect to take the philosophy of needs seriously or to draw on the theoretical resources of those theories to strengthen particular capabilitarian theories and applications.


24.4: Length Contraction
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As we saw in the examples from the previous section, time dilation implies “length contraction”. When an object is measured in a frame of reference that is at rest relative to the object, the length of the object, \(L\) , is called the “rest length” or the “proper length” of the object. If that object is moving relative to an observer, the observer will measure the object to be shorter, and have a “contracted length”, \(L'\) , given by:
\[L'=L\sqrt{a-\frac{v^{2}}{c^{2}}}=\frac{L}{\gamma}\]
In Example 24.3.2 , Alice measured a contracted distance between Earth and Proxima Centauri, as she was in a frame of reference that is moving relative to the Earth-Proxima Centauri reference frame. One point that is important to note is that length contraction only occurs along the direction parallel to the direction of motion.
Example \(\PageIndex{1}\)
A square painting hanging in a museum has a side with a length of \(1\text{m}\) . If you view the stationary painting from a train moving in the horizontal direction at a speed of \(0.85c\) , what is the surface area of the painting that you measure?
Solution
Since your train is moving horizontally, only the horizontal dimension of the painting will be contracted. The gamma factor is given by:
\[\begin{aligned} \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} =\frac{1}{\sqrt{1-0.85^2}}=1.9\end{aligned}\]
Thus, the horizontal side of the painting will have a contracted length:
\[\begin{aligned} L'=\frac{L}{\gamma}=\frac{(1\text{m})}{(1.9)}=0.53\text{m}\end{aligned}\]
The area of the painting, as measured in the moving frame of reference, is given by:
\[\begin{aligned} A= (1\text{m})(0.53\text{m})=0.53\text{m}^{2}\end{aligned}\]
Exercise \(\PageIndex{1}\)
What speed must an object travel in order for it to appear \(1\)% shorter
- \(0.01c\)
- \(0.14c\)
- \(0.99c\)
- \(0.65c\)
- Answer
-
B.
Olivia's Thoughts
In these types of problems, it is important to keep track of which reference frame gives us the proper length and which gives us the proper time. I recommend taking a minute at the beginning of each problem to think about this, and jot it down at the start of your solution. Take the example in which our friend Alice travels from Earth to Proxima Centuri on a spaceship. We are dealing with Earth's frame of reference, which I will call \(E\), and the Alice's frame of reference, which I will call \(A\). We are interested in the time elapsed on the spaceship and the distance to Proxima Centuri. The Earth is
moving
relative to the spaceship's clock and is
stationary
relative to Proxima Centuri. Thus, frame \(E\) gives us the
dilated time
, \(\Delta t'\), and the
proper length
, \(L\). Alice is
stationary
relative to the spaceship's clock and is
moving
relative to Proxima Centuri. Thus, frame \(A\) gives us the
proper time
, \(\Delta t\), and the
contracted length
, \(L'\).
At the top of my solution, I would write:
\[\begin{align*}
\textrm{Frame}\, E: L, \Delta t'\\[4pt]
\textrm{Frame}\, A: L', \Delta t
\end{align*}\]
To check that we got this right, the speed of the spaceship should not change depending on the reference frame. In the Earth's frame of reference, the velocity is given by \(v=L/\Delta t'\). In Alice's frame, it is given by \(v=L'/\Delta t\), or, using the gamma factor: \(v=(L/\gamma)/(\Delta t'/\gamma)=L/\Delta t'\), as expected.
Length contraction also allows us to discuss a famous paradox (the “barn”, or “ladder” or “barn-pole” paradox). Consider a train that has a rest length of \(500\text{m}\) , traveling at a speed such that \(\gamma = 2.5\) . As the train goes by, from Earth, it appears to have a (contracted) length:
\[\begin{aligned} L'_{train}=\frac{(500\text{m})}{2.5}=200\text{m}\end{aligned}\]
Suppose that there is a tunnel on Earth that is exactly \(200\text{m}\) long, so that the train, when contracted, will fit in the tunnel. When the train passes, an operator briefly closes (and re-opens) the doors at the ends of the tunnel, briefly “capturing” the train, and since the train is contracted, it never hits any of the doors, and all is fine.
From the train’s frame of reference, the train has a proper length of \(500\text{m}\) , and the tunnel is contracted to a length of:
\[\begin{aligned} L'_{tunnel}=\frac{(200\text{m})}{(2.5)}=80\text{m}\end{aligned}\]
Thus, from the train’s perspective, if the doors of the tunnel are closed, there is no way that the \(500\text{m}\) long train can ever fit in the \(80\text{m}\) long tunnel, as illustrated in Figure \(\PageIndex{1}\). So what happens when the operator on Earth closes the doors of the tunnel to briefly “capture” the train?
Clearly, people on the Earth and people on the train have to agree on whether the train was destroyed by the tunnel doors. The operator on Earth can clearly close both doors of the tunnel when the train is inside and not destroy the train. Hence, people on the train must agree that the train never collided with the doors, and that the doors were closed. The answer to this paradox lies in the fact that simultaneity is relative. The tunnel operator believes that she has closed the two doors of the tunnel at exactly the same time, precisely when the contracted train is lined up with the tunnel. However, to people on the train, in a different frame of reference, the doors did not close at the same time, since events that are simultaneous in one frame of reference are not necessarily simultaneous in a different frame of reference. To people on the train, there was never a time when the train was in the tunnel and both doors were closed at the same time!
Exercise \(\PageIndex{2}\)
Referring to the above paradox, to people on the train, which tunnel door closes first?
- The door at the entrance of the tunnel closes first.
- The door at the exit of the tunnel closes first.
- Answer
-
B.


11.10: Introduction
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Learning Outcomes
After reading this chapter, you should be able to answer these questions:
- What is the marketing concept and relationship-building?
- How do managers create a marketing strategy?
- What is the marketing mix?
- How do consumers and organizations make buying decisions?
- What are the five basic forms of consumer and business market segmentation?
- What is a product, and how is it classified?
- How do organizations create new products?
- What are the stages of the product life cycle?
- What strategies are used for pricing products, and what are the future trends?
- What trends are occurring in products and pricing?
EXPLORING BUSINESS CAREERS
Rachel Kuhr: Mark Cuban’s Shark Tank Empire
Rachel Kuhr is the product innovation and development specialist for Mark Cuban’s investments in the ABC show Shark Tank . Two years ago, after watching an episode, Kuhr e-mailed Mark Cuban and attached a resume that highlighted her mechanical engineering and product development expertise. Her approach appealed to Cuban, and she was contacted the next day by Abe Minkara, head of Cuban’s business development team. After a Skype interview in which Minkara was impressed with Kuhr’s skill set of both creativity and attention to process, she was hired to fill that role and work with several start-ups that Cuban acquired an investment in through the show.
Kuhr now coaches and collaborates with over 60 companies that reside in Cuban’s business portfolio. Rather than start out with detailed plans and building sophisticated prototypes, Kuhr favors using things like a whiteboard, Post-it notes, colored pens, and highlighters to sketch out the ideas. Such an approach uses the best practices from brainstorming that allow fatal flaws to steer the direction of product development before spending lots of resources, both human and financial, on a single idea for too long. This approach also allows the product development team to incorporate the user experience, which is sometimes overlooked when the focus is squarely on the product.
One of the companies that Kuhr works with created Chapul Cricket Bars. Chapul Cricket Bars was the first company to use insect-based “flour” in the manufacture of high- energy protein bars. After the deal on Shark Tank, company founder Pat Crowley and Kuhr decided to take the flying insect logo off the product design and renamed the bars with names such as Aztec, Matcha, and Chaco instead of the “Cricket Bar” name.
Another Cuban investment was the Austin, Texas–based BeatBox Beverages. To better understand how typical consumers would relate to boxed flavored cocktails, Kuhr attended several fraternity parties at Southern Methodist University and off-campus bars. She asked questions that addressed how a variety of consumers decide on what to drink on different occasions and in different settings. Since securing a $1 million dollar investment from Cuban, and working with Rachel Kuhr, online and distribution sales through stores has skyrocketed according to Justin Fenchel, BeatBox Beverages’ CEO.
Sources: Cheryl Hall, “Why Rachel Kuhr Is the Innovator for Mark Cuban’s Shark Tank Startups,” Dallas News , https://www.dallasnews.com , accessed October 1, 2017; “About Us,” https://chapul.com , accessed October 1, 2017; “The Story,” https://www.beatboxbeverages.com , accessed October 1, 2017; Teddy Nykiel, “Shark Tank’s Biggest Deals and How They Panned Out,” NerdWallet , https://www.nerdwallet.com , January 9, 2015.
Marketing plays a key role in the success of businesses. It is the task of marketing to generate sales for the firm. Sales revenue, in turn, pays workers’ salaries, buys supplies, covers the costs of new buildings and equipment, and hopefully enables the company to earn a profit. This chapter looks at the nature of marketing and the creation of product and pricing strategies to meet customers’ needs. In this chapter, you will learn about the marketing concept, marketing strategies, and consumer and business buying decisions. You will also see how the marketing mix is used to create sales opportunities. We discuss how new products are created and how they go through periods of sales growth and then decline. Next you will discover how managers set prices to reach organizational goals.


23.1: The Relationship Between Inflation and Unemployment
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- Boundless
- Boundless
The Phillips Curve
The Phillips curve shows the inverse relationship between inflation and unemployment: as unemployment decreases, inflation increases.
learning objectives
- Review the historical evidence regarding the theory of the Phillips curve
The Phillips curve relates the rate of inflation with the rate of unemployment. The Phillips curve argues that unemployment and inflation are inversely related: as levels of unemployment decrease, inflation increases. The relationship, however, is not linear. Graphically, the short-run Phillips curve traces an L-shape when the unemployment rate is on the x-axis and the inflation rate is on the y-axis.
Theoretical Phillips Curve : The Phillips curve shows the inverse trade-off between inflation and unemployment. As one increases, the other must decrease. In this image, an economy can either experience 3% unemployment at the cost of 6% of inflation, or increase unemployment to 5% to bring down the inflation levels to 2%.
History
The early idea for the Phillips curve was proposed in 1958 by economist A.W. Phillips. In his original paper, Phillips tracked wage changes and unemployment changes in Great Britain from 1861 to 1957, and found that there was a stable, inverse relationship between wages and unemployment. This correlation between wage changes and unemployment seemed to hold for Great Britain and for other industrial countries. In 1960, economists Paul Samuelson and Robert Solow expanded this work to reflect the relationship between inflation and unemployment. Because wages are the largest components of prices, inflation (rather than wage changes) could be inversely linked to unemployment.
The theory of the Phillips curve seemed stable and predictable. Data from the 1960’s modeled the trade-off between unemployment and inflation fairly well. The Phillips curve offered potential economic policy outcomes: fiscal and monetary policy could be used to achieve full employment at the cost of higher price levels, or to lower inflation at the cost of lowered employment. However, when governments attempted to use the Phillips curve to control unemployment and inflation, the relationship fell apart. Data from the 1970’s and onward did not follow the trend of the classic Phillips curve. For many years, both the rate of inflation and the rate of unemployment were higher than the Phillips curve would have predicted, a phenomenon known as “stagflation. ” Ultimately, the Phillips curve was proved to be unstable, and therefore, not usable for policy purposes.
US Phillips Curve (2000 – 2013) : The data points in this graph span every month from January 2000 until April 2013. They do not form the classic L-shape the short-run Phillips curve would predict. Although it was shown to be stable from the 1860’s until the 1960’s, the Phillips curve relationship became unstable – and unusable for policy-making – in the 1970’s.
The Relationship Between the Phillips Curve and AD-AD
Changes in aggregate demand cause movements along the Phillips curve, all other variables held constant.
learning objectives
- Relate aggregate demand to the Phillips curve
The Phillips Curve Related to Aggregate Demand
The Phillips curve shows the inverse trade-off between rates of inflation and rates of unemployment. If unemployment is high, inflation will be low; if unemployment is low, inflation will be high.
The Phillips curve and aggregate demand share similar components. The Phillips curve is the relationship between inflation, which affects the price level aspect of aggregate demand, and unemployment, which is dependent on the real output portion of aggregate demand. Consequently, it is not far-fetched to say that the Phillips curve and aggregate demand are actually closely related.
To see the connection more clearly, consider the example illustrated by. Let’s assume that aggregate supply, AS, is stationary, and that aggregate demand starts with the curve, AD 1 . There is an initial equilibrium price level and real GDP output at point A. Now, imagine there are increases in aggregate demand, causing the curve to shift right to curves AD 2 through AD4. As aggregate demand increases, unemployment decreases as more workers are hired, real GDP output increases, and the price level increases; this situation describes a demand-pull inflation scenario.
Phillips Curve and Aggregate Demand : As aggregate demand increases from AD1 to AD4, the price level and real GDP increases. This translates to corresponding movements along the Phillips curve as inflation increases and unemployment decreases.
As more workers are hired, unemployment decreases. Moreover, the price level increases, leading to increases in inflation. These two factors are captured as equivalent movements along the Phillips curve from points A to D. At the initial equilibrium point A in the aggregate demand and supply graph, there is a corresponding inflation rate and unemployment rate represented by point A in the Phillips curve graph. For every new equilibrium point (points B, C, and D) in the aggregate graph, there is a corresponding point in the Phillips curve. This illustrates an important point: changes in aggregate demand cause movements along the Phillips curve.
The Long-Run Phillips Curve
The long-run Phillips curve is a vertical line at the natural rate of unemployment, so inflation and unemployment are unrelated in the long run.
learning objectives
- Examine the NAIRU and its relationship to the long term Phillips curve
The Phillips curve shows the trade-off between inflation and unemployment, but how accurate is this relationship in the long run? According to economists, there can be no trade-off between inflation and unemployment in the long run. Decreases in unemployment can lead to increases in inflation, but only in the short run. In the long run, inflation and unemployment are unrelated. Graphically, this means the Phillips curve is vertical at the natural rate of unemployment, or the hypothetical unemployment rate if aggregate production is in the long-run level. Attempts to change unemployment rates only serve to move the economy up and down this vertical line.
Natural Rate Hypothesis
The natural rate of unemployment theory, also known as the non-accelerating inflation rate of unemployment (NAIRU) theory, was developed by economists Milton Friedman and Edmund Phelps. According to NAIRU theory, expansionary economic policies will create only temporary decreases in unemployment as the economy will adjust to the natural rate. Moreover, when unemployment is below the natural rate, inflation will accelerate. When unemployment is above the natural rate, inflation will decelerate. When the unemployment rate is equal to the natural rate, inflation is stable, or non-accelerating.
An Example
To get a better sense of the long-run Phillips curve, consider the example shown in. Assume the economy starts at point A and has an initial rate of unemployment and inflation rate. If the government decides to pursue expansionary economic policies, inflation will increase as aggregate demand shifts to the right. This is shown as a movement along the short-run Phillips curve, to point B, which is an unstable equilibrium. As aggregate demand increases, more workers will be hired by firms in order to produce more output to meet rising demand, and unemployment will decrease. However, due to the higher inflation, workers’ expectations of future inflation changes, which shifts the short-run Phillips curve to the right, from unstable equilibrium point B to the stable equilibrium point C. At point C, the rate of unemployment has increased back to its natural rate, but inflation remains higher than its initial level.
NAIRU and Phillips Curve : Although the economy starts with an initially low level of inflation at point A, attempts to decrease the unemployment rate are futile and only increase inflation to point C. The unemployment rate cannot fall below the natural rate of unemployment, or NAIRU, without increasing inflation in the long run.
The reason the short-run Phillips curve shifts is due to the changes in inflation expectations. Workers, who are assumed to be completely rational and informed, will recognize their nominal wages have not kept pace with inflation increases (the movement from A to B), so their real wages have been decreased. As such, in the future, they will renegotiate their nominal wages to reflect the higher expected inflation rate, in order to keep their real wages the same. As nominal wages increase, production costs for the supplier increase, which diminishes profits. As profits decline, suppliers will decrease output and employ fewer workers (the movement from B to C). Consequently, an attempt to decrease unemployment at the cost of higher inflation in the short run led to higher inflation and no change in unemployment in the long run.
The NAIRU theory was used to explain the stagflation phenomenon of the 1970’s, when the classic Phillips curve could not. According to the theory, the simultaneously high rates of unemployment and inflation could be explained because workers changed their inflation expectations, shifting the short-run Phillips curve, and increasing the prevailing rate of inflation in the economy. At the same time, unemployment rates were not affected, leading to high inflation and high unemployment.
The Short-Run Phillips Curve
The short-run Phillips curve depicts the inverse trade-off between inflation and unemployment.
learning objectives
- Interpret the short-run Phillips curve
The Phillips curve depicts the relationship between inflation and unemployment rates. The long-run Phillips curve is a vertical line that illustrates that there is no permanent trade-off between inflation and unemployment in the long run. However, the short-run Phillips curve is roughly L-shaped to reflect the initial inverse relationship between the two variables. As unemployment rates increase, inflation decreases; as unemployment rates decrease, inflation increases.
Short-Run Phillips Curve : The short-run Phillips curve shows that in the short-term there is a tradeoff between inflation and unemployment. Contrast it with the long-run Phillips curve (in red), which shows that over the long term, unemployment rate stays more or less steady regardless of inflation rate.
Consider the example shown in. When the unemployment rate is 2%, the corresponding inflation rate is 10%. As unemployment decreases to 1%, the inflation rate increases to 15%. On the other hand, when unemployment increases to 6%, the inflation rate drops to 2%.
Historical application
During the 1960’s, the Phillips curve rose to prominence because it seemed to accurately depict real-world macroeconomics. However, the stagflation of the 1970’s shattered any illusions that the Phillips curve was a stable and predictable policy tool. Nowadays, modern economists reject the idea of a stable Phillips curve, but they agree that there is a trade-off between inflation and unemployment in the short-run. Given a stationary aggregate supply curve, increases in aggregate demand create increases in real output. As output increases, unemployment decreases. With more people employed in the workforce, spending within the economy increases, and demand-pull inflation occurs, raising price levels.
Therefore, the short-run Phillips curve illustrates a real, inverse correlation between inflation and unemployment, but this relationship can only exist in the short run . The idea of a stable trade-off between inflation and unemployment in the long run has been disproved by economic history.
Relationship Between Expectations and Inflation
There are two theories of expectations (adaptive or rational) that predict how people will react to inflation.
learning objectives
- Distinguish adaptive expectations from rational expectations
The short-run Phillips curve is said to shift because of workers’ future inflation expectations. Yet, how are those expectations formed? There are two theories that explain how individuals predict future events.
Real versus Nominal Quantities
To fully appreciate theories of expectations, it is helpful to review the difference between real and nominal concepts. Anything that is nominal is a stated aspect. In contrast, anything that is real has been adjusted for inflation. To make the distinction clearer, consider this example. Suppose you are opening a savings account at a bank that promises a 5% interest rate. This is the nominal, or stated, interest rate. However, suppose inflation is at 3%. The real interest rate would only be 2% (the nominal 5% minus 3% to adjust for inflation).
The difference between real and nominal extends beyond interest rates. In an earlier atom, the difference between real GDP and nominal GDP was discussed. The distinction also applies to wages, income, and exchange rates, among other values.
Adaptive Expectations
The theory of adaptive expectations states that individuals will form future expectations based on past events. For example, if inflation was lower than expected in the past, individuals will change their expectations and anticipate future inflation to be lower than expected.
To connect this to the Phillips curve, consider. Assume the economy starts at point A at the natural rate of unemployment with an initial inflation rate of 2%, which has been constant for the past few years. Accordingly, because of the adaptive expectations theory, workers will expect the 2% inflation rate to continue, so they will incorporate this expected increase into future labor bargaining agreements. This way, their nominal wages will keep up with inflation, and their real wages will stay the same.
Expectations and the Phillips Curve : According to adaptive expectations theory, policies designed to lower unemployment will move the economy from point A through point B, a transition period when unemployment is temporarily lowered at the cost of higher inflation. However, eventually, the economy will move back to the natural rate of unemployment at point C, which produces a net effect of only increasing the inflation rate.According to rational expectations theory, policies designed to lower unemployment will move the economy directly from point A to point C. The transition at point B does not exist as workers are able to anticipate increased inflation and adjust their wage demands accordingly.
Now assume that the government wants to lower the unemployment rate. To do so, it engages in expansionary economic activities and increases aggregate demand. As aggregate demand increases, inflation increases. Because of the higher inflation, the real wages workers receive have decreased. For example, assume each worker receives $100, plus the 2% inflation adjustment. Each worker will make $102 in nominal wages, but $100 in real wages. Now, if the inflation level has risen to 6%. Workers will make $102 in nominal wages, but this is only $96.23 in real wages.
Although the workers’ real purchasing power declines, employers are now able to hire labor for a cheaper real cost. Consequently, employers hire more workers to produce more output, lowering the unemployment rate and increasing real GDP. On, the economy moves from point A to point B.
However, workers eventually realize that inflation has grown faster than expected, their nominal wages have not kept pace, and their real wages have been diminished. They demand a 4% increase in wages to increase their real purchasing power to previous levels, which raises labor costs for employers. As labor costs increase, profits decrease, and some workers are let go, increasing the unemployment rate. Graphically, the economy moves from point B to point C.
This example highlights how the theory of adaptive expectations predicts that there are no long-run trade-offs between unemployment and inflation. In the short run, it is possible to lower unemployment at the cost of higher inflation, but, eventually, worker expectations will catch up, and the economy will correct itself to the natural rate of unemployment with higher inflation.
Rational Expectations
The theory of rational expectations states that individuals will form future expectations based on all available information, with the result that future predictions will be very close to the market equilibrium. For example, assume that inflation was lower than expected in the past. Individuals will take this past information and current information, such as the current inflation rate and current economic policies, to predict future inflation rates.
As an example of how this applies to the Phillips curve, consider again. Assume the economy starts at point A, with an initial inflation rate of 2% and the natural rate of unemployment. However, under rational expectations theory, workers are intelligent and fully aware of past and present economic variables and change their expectations accordingly. They will be able to anticipate increases in aggregate demand and the accompanying increases in inflation. As such, they will raise their nominal wage demands to match the forecasted inflation, and they will not have an adjustment period when their real wages are lower than their nominal wages. Graphically, they will move seamlessly from point A to point C, without transitioning to point B.
In essence, rational expectations theory predicts that attempts to change the unemployment rate will be automatically undermined by rational workers. They can act rationally to protect their interests, which cancels out the intended economic policy effects. Efforts to lower unemployment only raise inflation.
Shifting the Phillips Curve with a Supply Shock
Aggregate supply shocks, such as increases in the costs of resources, can cause the Phillips curve to shift.
learning objectives
- Give examples of aggregate supply shock that shift the Phillips curve
The Phillips curve shows the relationship between inflation and unemployment. In the short-run, inflation and unemployment are inversely related; as one quantity increases, the other decreases. In the long-run, there is no trade-off. In the 1960’s, economists believed that the short-run Phillips curve was stable. By the 1970’s, economic events dashed the idea of a predictable Phillips curve. What could have happened in the 1970’s to ruin an entire theory? Stagflation caused by a aggregate supply shock.
Stagflation and Aggregate Supply Shocks
Stagflation is a combination of the words “stagnant” and “inflation,” which are the characteristics of an economy experiencing stagflation: stagnating economic growth and high unemployment with simultaneously high inflation. The stagflation of the 1970’s was caused by a series of aggregate supply shocks. In this case, huge increases in oil prices by the Organization of Petroleum Exporting Countries (OPEC) created a severe negative supply shock. The increased oil prices represented greatly increased resource prices for other goods, which decreased aggregate supply and shifted the curve to the left. As aggregate supply decreased, real GDP output decreased, which increased unemployment, and price level increased; in other words, the shift in aggregate supply created cost-push inflation.
Aggregate Supply Shock : In this example of a negative supply shock, aggregate supply decreases and shifts to the left. The resulting decrease in output and increase in inflation can cause the situation known as stagflation.
Shifting the Phillips Curve
The aggregate supply shocks caused by the rising price of oil created simultaneously high unemployment and high inflation. At the time, the dominant school of economic thought believed inflation and unemployment to be mutually exclusive; it was not possible to have high levels of both within an economy. Consequently, the Phillips curve could not model this situation. For high levels of unemployment, there were now corresponding levels of inflation that were higher than the Phillips curve predicted; the Phillips curve had shifted upwards and to the right. Thus, the Phillips curve no longer represented a predictable trade-off between unemployment and inflation.
Disinflation
Disinflation is a decline in the rate of inflation, and can be caused by declines in the money supply or recessions in the business cycle.
learning objectives
- Identify situations with disinflation
Inflation is the persistent rise in the general price level of goods and services. Disinflation is a decline in the rate of inflation; it is a slowdown in the rise in price level. As an example, assume inflation in an economy grows from 2% to 6% in Year 1, for a growth rate of four percentage points. In Year 2, inflation grows from 6% to 8%, which is a growth rate of only two percentage points. The economy is experiencing disinflation because inflation did not increase as quickly in Year 2 as it did in Year 1, but the general price level is still rising. Disinflation is not to be confused with deflation, which is a decrease in the general price level.
Causes
Disinflation can be caused by decreases in the supply of money available in an economy. It can also be caused by contractions in the business cycle, otherwise known as recessions. The Phillips curve can illustrate this last point more closely. Consider an economy initially at point A on the long-run Phillips curve in. Suppose that during a recession, the rate that aggregate demand increases relative to increases in aggregate supply declines. This reduces price levels, which diminishes supplier profits. As profits decline, employers lay off employees, and unemployment rises, which moves the economy from point A to point B on the graph. Eventually, though, firms and workers adjust their inflation expectations, and firms experience profits once again. As profits increase, employment also increases, returning the unemployment rate to the natural rate as the economy moves from point B to point C. The expected rate of inflation has also decreased due to different inflation expectations, resulting in a shift of the short-run Phillips curve.
Disinflation : Disinflation can be illustrated as movements along the short-run and long-run Phillips curves.
Inflation vs. Deflation vs. Disinflation
To illustrate the differences between inflation, deflation, and disinflation, consider the following example. Assume the following annual price levels as compared to the prices in year 1:
- Year 1: 100% of Year 1 prices
- Year 2: 104% of Year 1 prices
- Year 3: 106% of Year 1 prices
- Year 4: 107% of Year 1 prices
- Year 5: 105% of Year 1 prices
As the economy moves through Year 1 to Year 4, there is a continued growth in the price level. This is an example of inflation; the price level is continually rising. However, between Year 2 and Year 4, the rise in price levels slows down. Between Year 2 and Year 3, the price level only increases by two percentage points, which is lower than the four percentage point increase between Years 1 and 2. The trend continues between Years 3 and 4, where there is only a one percentage point increase. This is an example of disinflation; the overall price level is rising, but it is doing so at a slower rate.
Between Years 4 and 5, the price level does not increase, but decreases by two percentage points. This is an example of deflation; the price rise of previous years has reversed itself.
Key Points
- The relationship between inflation rates and unemployment rates is inverse. Graphically, this means the short-run Phillips curve is L-shaped.
- A.W. Phillips published his observations about the inverse correlation between wage changes and unemployment in Great Britain in 1958. This relationship was found to hold true for other industrial countries, as well.
- From 1861 until the late 1960’s, the Phillips curve predicted rates of inflation and rates of unemployment. However, from the 1970’s and 1980’s onward, rates of inflation and unemployment differed from the Phillips curve’s prediction. The relationship between the two variables became unstable.
- Aggregate demand and the Phillips curve share similar components. The rate of unemployment and rate of inflation found in the Phillips curve correspond to the real GDP and price level of aggregate demand.
- Changes in aggregate demand translate as movements along the Phillips curve.
- If there is an increase in aggregate demand, such as what is experienced during demand-pull inflation, there will be an upward movement along the Phillips curve. As aggregate demand increases, real GDP and price level increase, which lowers the unemployment rate and increases inflation.
- The natural rate of unemployment is the hypothetical level of unemployment the economy would experience if aggregate production were in the long-run state.
- The natural rate hypothesis, or the non-accelerating inflation rate of unemployment (NAIRU) theory, predicts that inflation is stable only when unemployment is equal to the natural rate of unemployment. If unemployment is below (above) its natural rate, inflation will accelerate (decelerate).
- Expansionary efforts to decrease unemployment below the natural rate of unemployment will result in inflation. This changes the inflation expectations of workers, who will adjust their nominal wages to meet these expectations in the future. This leads to shifts in the short-run Phillips curve.
- The natural rate hypothesis was used to give reasons for stagflation, a phenomenon that the classic Phillips curve could not explain.
- The long-run Phillips curve is a vertical line at the natural rate of unemployment, but the short-run Phillips curve is roughly L-shaped.
- The inverse relationship shown by the short-run Phillips curve only exists in the short-run; there is no trade-off between inflation and unemployment in the long run.
- Economic events of the 1970’s disproved the idea of a permanently stable trade-off between unemployment and inflation.
- Nominal quantities are simply stated values. Real quantities are nominal ones that have been adjusted for inflation.
- Adaptive expectations theory says that people use past information as the best predictor of future events. If inflation was higher than normal in the past, people will expect it to be higher than anticipated in the future.
- Rational expectations theory says that people use all available information, past and current, to predict future events. If inflation was higher than normal in the past, people will take that into consideration, along with current economic indicators, to anticipate its future performance.
- According to adaptive expectations, attempts to reduce unemployment will result in temporary adjustments along the short-run Phillips curve, but will revert to the natural rate of unemployment. According to rational expectations, attempts to reduce unemployment will only result in higher inflation.
- In the 1970’s soaring oil prices increased resource costs for suppliers, which decreased aggregate supply. The resulting cost-push inflation situation led to high unemployment and high inflation ( stagflation ), which shifted the Phillips curve upwards and to the right.
- Stagflation is a situation where economic growth is slow (reducing employment levels) but inflation is high.
- The Phillips curve was thought to represent a fixed and stable trade-off between unemployment and inflation, but the supply shocks of the 1970’s caused the Phillips curve to shift. This ruined its reputation as a predictable relationship.
- Disinflation is not the same as deflation, when inflation drops below zero.
- During periods of disinflation, the general price level is still increasing, but it is occurring slower than before.
- The short-run and long-run Phillips curve may be used to illustrate disinflation.
Key Terms
- Phillips curve : A graph that shows the inverse relationship between the rate of unemployment and the rate of inflation in an economy.
- stagflation : Inflation accompanied by stagnant growth, unemployment, or recession.
- aggregate demand : The the total demand for final goods and services in the economy at a given time and price level.
- Natural Rate of Unemployment : The hypothetical unemployment rate consistent with aggregate production being at the long-run level.
- non-accelerating inflation rate of unemployment : (NAIRU); theory that describes how the short-run Phillips curve shifts in the long run as expectations change.
- Phillips curve : A graph that shows the inverse relationship between the rate of unemployment and the rate of inflation in an economy.
- adaptive expectations theory : A hypothesized process by which people form their expectations about what will happen in the future based on what has happened in the past.
- rational expectations theory : A hypothesized process by which people form their expectations about what will happen in the future based on all relevant information.
- supply shock : An event that suddenly changes the price of a commodity or service. It may be caused by a sudden increase or decrease in the supply of a particular good.
- disinflation : A decrease in the inflation rate.
- inflation : An increase in the general level of prices or in the cost of living.
- deflation : A decrease in the general price level, that is, in the nominal cost of goods and services.
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- Boundless. Provided by : Boundless Learning. Located at : http://www.boundless.com . License : CC BY-SA: Attribution-ShareAlike
- NAIRU-SR-and-LR. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:NAIRU-SR-and-LR.svg . License : Public Domain: No Known Copyright
- PhilCurve.png. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/File:PhilCurve.png . License : CC BY-SA: Attribution-ShareAlike
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- Economics supply shock. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Ec...pply_shock.png . License : Public Domain: No Known Copyright
- Provided by : lh5.googleusercontent.com/-B...inflation2.png. Located at : https://lh5.googleusercontent.com/-Bc5Yt-QMGXA/Uo3sjZ7SgxI/AAAAAAAAAXQ/1MksRdza_rA/s512/Phillipscurve_disinflation2.png . License : CC BY-SA: Attribution-ShareAlike


<IP_ADDRESS>: A Dialogue between Old England and New
Alas, dear Mother, fairest Queen and best,
With honour, wealth, and peace happy and blest,
What ails thee hang thy head, and cross thine arms,
And sit i' the dust to sigh these sad alarms?
What deluge of new woes thus over-whelm
The glories of thy ever famous Realm?
What means this wailing tone, this mournful guise?
Ah, tell thy Daughter; she may sympathize.

And thou a child, a Limb, and dost not feel
My weak'ned fainting body now to reel?
This physic-purging-potion I have taken
Will bring Consumption or an Ague quaking,
Unless some Cordial thou fetch from high,
Which present help may ease my malady.
If I decease, dost think thou shalt survive?
Or by my wasting state dost think to thrive?
Then weigh our case, if 't be not justly sad.
Let me lament alone, while thou art glad.
New England.And thus, alas, your state you much deplore
In general terms, but will not say wherefore.
What Medicine shall I seek to cure this woe,
If th' wound's so dangerous, I may not know?
But you, perhaps, would have me guess it out.
What, hath some Hengist like that Saxon stout
By fraud and force usurp'd thy flow'ring crown,
Or by tempestuous Wars thy fields trod down?
Or hath Canutus , that brave valiant Dane ,
The regal peaceful Sceptre from thee ta'en?

Or is 't intestine Wars that thus offend?
Do Maud and Stephen for the Crown contend?
Do Barons rise and side against their King,
And call in Foreign aid to help the thing?
Must Edward be depos'd? Or is 't the hour
That second Richard must be clapp'd i' th' Tower?
Or is it the fatal jar, again begun,
That from the red, white pricking Roses sprung?
Must Richmond's aid the Nobles now implore
To come and break the tushes of the Boar?
If none of these, dear Mother, what's your woe?
Pray, do not fear Spain's bragging Armado.
Doth your Ally, fair France , conspire your wrack,
Or doth the Scots play false behind your back?
Doth Holland quit you ill for all your love?
Whence is this storm, from Earth or Heaven above?
Is 't drought, is 't Famine, or is 't Pestilence?
Dost feel the smart, or fear the consequence?

Though Arms nor Purse she hath for your relief—
Such is her poverty,—yet shall be found
A suppliant for your help, as she is bound.
Old England.I must confess some of those Sores you name
My beauteous Body at this present maim,
But foreign Foe nor feigned friend I fear,
For they have work enough, thou knowest, elsewhere.
Nor is it Alcie's son and Henry's Daughter
Whose proud contention cause this slaughter;
Nor Nobles siding to make John no King,
French Louis unjustly to the Crown to bring;
No Edward , Richard , to lose rule and life,
Nor no Lancastrians to renew old strife;
No Crook-backt Tyrant now usurps the Seat,
Whose tearing tusks did wound, and kill, and threat.
No Duke of York nor Earl of March to soil
Their hands in Kindred's blood whom they did foil;
No need of Tudor Roses to unite:
None knows which is the Red or which the White.
Spain's braving Fleet a second time is sunk.
France knows how of my fury she hath drunk
By Edward third and Henry fifth of fame;
Her Lilies in my Arms avouch the same.
My Sister Scotland hurts me now no more,
Though she hath been injurious heretofore.
What Holland is, I am in some suspense,
But trust not much unto his Excellence.
For wants, sure some I feel, but more I fear;
And for the Pestilence, who knows how near?
Famine and Plague, two sisters of the Sword,
Destruction to a Land doth soon afford.
They're for my punishments ordain'd on high,
Unless thy tears prevent it speedily.
But yet I answer not what you demand
To shew the grievance of my troubled Land.
Before I tell the effect I'll shew the cause,
Which are my sins—the breach of sacred Laws:
Idolatry, supplanter of a Nation,
With foolish superstitious adoration,
Are lik'd and countenanc'd by men of might,
The Gospel is trod down and hath no right.
Church Offices are sold and bought for gain
That Pope had hope to find Rome here again.
For Oaths and Blasphemies did ever ear
From Beelzebub himself such language hear?
What scorning of the Saints of the most high!
What injuries did daily on them lie!
What false reports, what nick-names did they take,
Not for their own, but for their Master's sake!
And thou, poor soul, wast jeer'd among the rest;
Thy flying for the Truth I made a jest.
For Sabbath-breaking and for Drunkenness
Did ever Land profaneness more express?
From crying bloods yet cleansed am not I,
Martyrs and others dying causelessly.
How many Princely heads on blocks laid down
For nought but title to a fading Crown!

Because of Royal Stem, that was thy crime.
For Bribery, Adultery, for Thefts, and Lies
Where is the Nation I can't paralyze?
With Usury, Extortion, and Oppression,
These be the Hydras of my stout transgression;
These be the bitter fountains, heads, and roots
Whence flow'd the source, the sprigs, the boughs, and fruits.
Of more than thou canst hear or I relate,
That with high hand I still did perpetrate,
For these were threat'ned the woeful day
I mocked the Preachers, put it fair away.
The Sermons yet upon record do stand
That cried destruction to my wicked Land.
These Prophets' mouths (all the while) was stopt,
Unworthily, some backs whipt, and ears crept;
Their reverent cheeks bear the glorious marks
Of stinking, stigmatizing Romish Clerks;
Some lost their livings, some in prison pent,
Some grossly fined, from friends to exile went:
Their silent tongues to heaven did vengeance cry,
Who heard their cause, and wrongs judg'd righteously,
And will repay it sevenfold in my lap.
This is fore-runner of my after-clap.
Nor took I warning by my neighbors' falls.
I saw sad Germany's dismantled walls,
I saw her people famish'd, Nobles slain,
Her fruitful land a barren heath remain.
I saw (unmov'd) her Armies foil'd and fled,
Wives forc'd, babes toss'd, her houses calcined.
I saw strong Rochelle yield'd to her foe,
Thousands of starved Christians there also.
I saw poor Ireland bleeding out her last,
Such cruelty as all reports have past.
Mine heart obdurate stood not yet aghast.
Now sip I of that cup, and just 't may be
The bottom dregs reserved are for me.
New England.To all you've said, sad mother, I assent.
Your fearful sins great cause there 's to lament.

But all you say amounts to this effect,
Not what you feel, but what you do expect.
Pray, in plain terms, what is your present grief?
Then let's join heads and hands for your relief.
Old England.Well, to the matter, then. There's grown of late
'Twixt King and Peers a question of state:
Which is the chief, the law, or else the King?
One saith, it's he; the other, no such thing.
My better part in Court of Parliament
To ease my groaning land shew their intent
To crush the proud, and right to each man deal,
To help the Church, and stay the Common-Weal.
So many obstacles comes in their way
As puts me to a stand what I should say.
Old customs, new Prerogatives stood on.
Had they not held law fast, all had been gone,
Which by their prudence stood them in such stead
They took high Strafford lower by the head,
And to their Laud be 't spoke they held 'n th' Tower
All England's metropolitan that hour.
This done, an Act they would have passed fain
No prelate should his Bishopric retain.
Here tugg'd they hard indeed, for all men saw
This must be done by Gospel, not by law.
Next the Militia they urged sore.
This was denied, I need not say wherefore.
The King, displeased, at York himself absents.
They humbly beg return, shew their intents.
The writing, printing, posting to and fro,
Shews all was done; I'll therefore let it go.
But now I come to speak of my disaster.
Contention's grown 'twixt Subjects and their Master,
They worded it so long they fell to blows,
That thousands lay on heaps. Here bleeds my woes.
I that no wars so many years have known
Am now destroy'd and slaughter'd by mine own.
But could the field alone this strife decide,
One battle, two, or three I might abide,
But these may be beginnings of more woe—
Who knows, the worst, the best may overthrow!
Religion, Gospel, here lies at the stake,
Pray now, dear child, for sacred Zion's sake,
Oh, pity me in this sad perturbation,
My plundered Towns, my houses' devastation,
My ravisht virgins, and my young men slain,
My wealthy trading fallen, my dearth of grain.
The seedtime's come, but Ploughman hath no hope
Because he knows not who shall inn his crop.
The poor they want their pay, their children bread,
Their woful mothers' tears unpitied.
If any pity in thy heart remain,
Or any child-like love thou dost retain,
For my relief now use thy utmost skill,
And recompense me good for all my ill.
New England.Dear mother, cease complaints, and wipe your eyes,
Shake off your dust, cheer up, and now arise.

Your griefs I pity much but should do wrong,
To weep for that we both have pray'd for long,
To see these latter days of hop'd-for good,
That Right may have its right, though 't be with blood.
After dark Popery the day did clear;
But now the Sun in's brightness shall appear.
Blest be the Nobles of thy Noble Land
With (ventur'd lives) for truth's defence that stand.
Blest be thy Commons, who for Common good
And thy infringed Laws have boldly stood.
Blest be thy Counties, who do aid thee still
With hearts and states to testify their will.
Blest be thy Preachers, who do cheer thee on.
Oh, cry: the sword of God and Gideon !
And shall I not on them wish Mero's curse
That help thee not with prayers, arms, and purse?
And for my self, let miseries abound
If mindless of thy state I e'er be found.
These are the days the Church's foes to crush,
To root out Prelates, head, tail, branch, and rush.

Go on, brave Essex, shew whose son thou art,
Not false to King, nor Country in thy heart,
But those that hurt his people and his Crown,
By force expel, destroy, and tread them down.
Let Gaols be fill'd with th' remnant of that pack,
And sturdy Tyburn loaded till it crack.
And ye brave Nobles, chase away all fear,
And to this blessed Cause closely adhere.
O mother, can you weep and have such Peers?
When they are gone, then drown your self in tears,
If now you weep so much, that then no more
The briny Ocean will o'erflow your shore.
These, these are they (I trust) with Charles our king,
Out of all mists such glorious days will bring
That dazzled eyes, beholding, much shall wonder
At that thy settled Peace, thy wealth, and splendour,
Thy Church and Weal establish'd in such manner
That all shall joy that thou display'dst thy banner,
And discipline erected so, I trust,
That nursing Kings shall come and lick thy dust.
Then Justice shall in all thy Courts take place
Without respect of persons or of case.
Then bribes shall cease, and suits shall not stick long,
Patience and purse of Clients for to wrong.
Then High Commissions shall fall to decay,
And Pursuivants and Catchpoles want their pay.
So shall thy happy Nation ever flourish,
When truth and righteousness they thus shall nourish.
When thus in Peace, thine Armies brave send out
To sack proud Rome , and all her vassals rout.
There let thy name, thy fame, and valour shine,
As did thine Ancestors' in Palestine ,
And let her spoils full pay with int'rest be
Of what unjustly once she poll'd from thee.
Of all the woes thou canst let her be sped,
Execute to th' full the vengeance threatened.
Bring forth the beast that rul'd the world with's beck,
And tear his flesh, and set your feet on's neck,
And make his filthy den so desolate
To th' 'stonishment of all that knew his state.
This done, with brandish'd swords to Turkey go,—
(For then what is it but English blades dare do?)
And lay her waste, for so's the sacred doom,
And do to Gog as thou hast done to Rome .
Oh Abraham's seed, lift up your heads on high,
For sure the day of your redemption's nigh.
The scales shall fall from your long blinded eyes,
And him you shall adore who now despise.
Then fullness of the Nations in shall flow,
And Jew and Gentile to one worship go.
Then follows days of happiness and rest.
Whose lot doth fall to live therein is blest.
No Canaanite shall then be found 'n th' land,
And holiness on horses' bells shall stand.
If this make way thereto, then sigh no more,
But if at all thou didst not see 't before.
Farewell, dear mother; Parliament, prevail,
And in a while you'll tell another tale.


5.11: Introduction to Finding Qualified Job Applicants
Recruiting is the art of attraction, a process that requires a clear understanding of what makes the company unique as well as what type of person a company wants to attract. Recruiting is often a process of discovery and evaluation for both company and candidate.
The reality is there’s no one best place to work. In the retail space alone, candidates can choose from cult brands including Apple, IKEA, and Lululemon as well as a number of beloved regional brands. With so many “Best of” and “Great Place to Work” options and the unemployment rate at historic lows, the market for talent is competitive.
Read More
Take a look at these reviews on great places to work:
- Great Places to Work
- Business Insider’s “ The 10 best retail companies to work for in 2017, according to employees ”
In this section, we’ll discuss effective strategies for identifying, attracting and recruiting qualified candidates, including equal opportunity laws to be aware of throughout the hiring process.
- Introduction to Finding Qualified Job Applicants. Authored by : Nina Burokas. Provided by : Lumen Learning. License : CC BY: Attribution


6.4: Conditional Probability
Learning Objectives
In this section, you will learn to:
- Recognize situations involving conditional probability.
- Calculate conditional probabilities.
Prerequisite Skills
Before you get started, take this prerequisite quiz.
1. Three coins are tossed. Write a sample space for the experiment:
a. If all three coins are pennies.
b. If the three coins are a penny, a nickel, and a dime.
- Click here to check your answer
-
a. {3H, 2H1T, 1H2T, 3T} or {HHH, HHT, HTT, TTT}
b. {HHH, HHT, HTH, HTT, TTT, TTH, THT, THH}
If you missed this problem, review Section 6.1 . (Note that this will open in a new window.)
2. Refer to the following table, which shows the distribution of Democratic and Republican U.S by gender in the 114 th Congress as of January 2015.
|
MALE(M) |
FEMALE(F) |
TOTAL |
|
|
DEMOCRATS (D) |
30 |
14 |
44 |
|
REPUBLICANS(R) |
48 |
6 |
54 |
|
OTHER (T) |
2 |
0 |
2 |
|
TOTALS |
80 |
20 |
100 |
Use this table to determine the following probabilities.
a. \(P(M \cap D)\)
b. \(P(F \cap R)\)
- Click here to check your answer
-
a. \(P(M \cap D)=\dfrac{3}{10}\)
b. \(P(F \cap R)=\dfrac{3}{50}\)
If you missed this problem, review Section 6.2 . (Note that this will open in a new window.)
3. Calculate each expression:
a. \(10P2\)
b. \(15P6\)
- Click here to check your answer
-
a. \(10P2=90\)
b. \(15P6=3,603,600\)
If you missed this problem, review Section 5.3 . (Note that this will open in a new window.)
4. Without using a calculator, simplify \(\dfrac{a}{b}\) if \(a=\dfrac{2}{15}\) and \(b=\dfrac{4}{15}\).
- Click here to check your answer
-
\(b=\dfrac{1}{2}\)
If you missed this problem, review here . (Note that this will open a different textbook in a new window.)
Suppose a friend, an avid traveler, asks you for the probability that they will encounter snow today.
If your friend is in Alaska today, the probability of snow today might be quite substantial. If they are in Florida though, the probability of snow today is likely very tiny.
Let:
- \(\mathrm{S}\) = the event that it will snow today
- \(\mathrm{A}\) = the event that your friend is in Alaska today
- \(\mathrm{F}\) = the event that your friend is in Florida today
Because the probability of snow is affected by the location, we can’t just write \(\mathrm{P(S)}\) for the probability of snow. We need to indicate the other information we know. We need to use conditional probability .
The event we are interested in is event \(\mathrm{S}\) for snow. The other event is called the condition, representing the location in this case.
We represent conditional probability using a vertical line | that means “if”, or “given that”, or “if we know that”. The event of interest appears on the left of the |. The condition appears on the right side of the |. This will appear as \(\mathbf{P} \left( \mathbf{event} | \mathbf{condition}\right) \)
The probability it will snow given that (if) your friend is in Alaska is represented by \(\mathbf{P} \left( \mathbf{S} | \mathbf{A}\right) \). In this case, the condition is \(\mathrm{A}\).
The probability that it will snow given that (if) your friend is in Florida is represented by \(\mathbf{P}(\mathbf{S} | \mathbf{F})\). In this case, the condition is \(\mathrm{F}\).
Now, let’s examine a situation where we can calculate some probabilities.
Suppose you and a friend play a game that involves choosing a single card from a well-shuffled deck.  Should you play the game?
You reason in the following manner. Since there are four kings in the deck, the probability of obtaining a king is 4/52 or 1/13. So, probability of not obtaining a king is 12/13. This implies that the ratio of your winning to losing is 1 to 12, while the payoff ratio is only $1 to $5. Therefore, you determine that you should not play.
But consider the following scenario. While your friend was dealing the card, you happened to get a glance of it and noticed that the card was a face card.
Since there are 12 face cards in the deck, the total elements in the sample space are no longer 52, but just 12. This means the chance of obtaining a king is 4/12 or 1/3. So your chance of winning is 1/3 and of losing 2/3. This makes your winning to losing ratio 1 to 2 which fares much better with the payoff ratio of $1 to $5. This time, you determine that you should play.
In the second part of the above example, we were finding the probability of obtaining a king knowing that a face card had shown. This is an example of conditional probability . Whenever we are finding the probability of an event \(\mathrm{E}\) under the condition that another event \(\mathrm{F}\) has happened, we are finding conditional probability.
The symbol \(\mathrm{P(E | F)}\) denotes the problem of finding the probability of \(\mathrm{E}\) given that \(\mathrm{F}\) has occurred. We read \(\mathrm{P(E | F)}\) as "the probability of \(\mathrm{E}\), given \(\mathrm{F}\)."
Example \(\PageIndex{1}\)
A family has three children.
- Find the probability of having two boys and one girl.
- Find the probability of having two boys and one girl given that the first born is a boy.
Solution
First, we the sample space for a family of three children as follows.
\[S = \{BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG\} \nonumber \]
Let event \(\mathrm{E}\) be that the family has two boys and a girl, and \(\mathrm{F}\) that the first born is a boy.
a. \(\mathrm{P(E)} = 3/8\) because 3 of the 8 options in the sample space include two boys and one girl.
b. Given that the first born is a boy though, our possibilities narrow down to four outcomes: BBB, BBG, BGB, and BGG.
Among the four, BBG and BGB represent two boys and a girl.
Therefore, \(\mathrm{P(E | F)}\) = 2/4 or 1/2.
Example \(\PageIndex{2}\)
One six sided die is rolled once.
- Find the probability that the result is even.
- Find the probability that the result is even given that the result is greater than three.
Solution
The sample space is \(\mathrm{S} = {1,2,3,4,5,6}\)
Let event \(\mathrm{E}\) be that the result is even and \(\mathrm{T}\) be that the result is greater than 3.
a. \(\mathrm{P(E)}\) = 3/6 because \(\mathrm{E} = {2,4,6}\)
b. Because \(T = {4,5,6}\), we know that 1, 2, 3 cannot occur; only outcomes 4, 5, 6 are possible. Therefore of the values in \(\mathrm{E}\), only 4, 6 are possible.
Therefore, \(\mathrm{P(E|T)}\) = 2/3
Example \(\PageIndex{3}\)
A fair coin is tossed twice.
- Find the probability that the result is is two heads.
- Find the probability that the result is two heads given that at least one head is obtained.
Solution
The sample space is \(S = {HH, HT, TH, TT}\)
Let event \(\mathrm{E}\) be that the two heads are obtained and \(\mathrm{F}\) be at least one head is obtained
a. \(\mathrm{P(E)}\) = 1/4 because \(\mathrm{E} = {HH}\) and the sample space \(\mathrm{S}\) has 4 outcomes.
b. \(\mathrm{F} = {HH, HT, TH}\). Since at least one head was obtained, TT did not occur.
We are interested in the probability event \(\mathrm{E}={HH}\) out of the 3 outcomes in the reduced sample space F.
Therefore, \(\mathrm{P(E|F)}\) = 1/3
Let us now develop a formula for the conditional probability \(\mathrm{P(E | F)}\).
Suppose an experiment consists of \(n\) equally likely events. Further suppose that there are \(m\) elements in \(\mathrm{F}\), and \(c\) elements in \(\mathrm{E} \cap \mathrm{F}\), as shown in the following Venn diagram.
If the event \(\mathrm{F}\) has occurred, the set of all possible outcomes is no longer the entire sample space, but instead, the subset \(\mathrm{F}\). Therefore, we only look at the set \(\mathrm{F}\) and at nothing outside of \(\mathrm{F}\). Since \(\mathrm{F}\) has \(m\) elements, the denominator in the calculation of \(\mathrm{P(E | F)}\) is \(m\). We may think that the numerator for our conditional probability is the number of elements in \(\mathrm{E}\). But clearly we cannot consider the elements of \(\mathrm{E}\) that are not in \(\mathrm{F}\). We can only count the elements of \(\mathrm{E}\) that are in \(\mathrm{F}\), that is, the elements in \(\mathrm{E} \cap \mathrm{F}\). Therefore,
\[\mathrm{P}(\mathrm{E} | \mathrm{F})=\frac{\mathrm{c}}{\mathrm{m}} \nonumber\]
Dividing both the numerator and the denominator by \(n\), we get
\[\mathrm{P(E | F)}=\frac{c / n}{m / n} \nonumber\]
But \(c/n = \mathrm{P}(\mathrm{E} \cap \mathrm{F})\), and \(m/n = \mathrm{P}(\mathrm{F})\).
Substituting, we derive the following formula for \(\mathrm{P}(\mathrm{E} | \mathrm{F})\).
Conditional Probability Rule
For two events \(\mathrm{E}\) and \(\mathrm{F}\), the probability of "\(\mathrm{E}\) Given \(\mathrm{F}\)" is
\[\mathbf{P}(\mathbf{E} | \mathbf{F})=\frac{\mathbf{P}(\mathbf{E} \cap \mathbf{F})}{\mathbf{P}(\mathbf{F})} \]
Example \(\PageIndex{4}\)
A single die is rolled. Use the above formula to find the conditional probability of obtaining an even number given that a number greater than three has shown.
Solution
Let \(\mathrm{E}\) be the event that an even number shows, and \(\mathrm{F}\) be the event that a number greater than three shows. We want \(\mathrm{P}(\mathrm{E} | \mathrm{F})\).
\(\mathrm{E} = {2, 4, 6}\) and \(\mathrm{F} = {4, 5, 6}\). Which implies, \(\mathrm{E} \cap \mathrm{F} = { 4, 6}\)
Therefore, \(\mathrm{P}(\mathrm{F})\) = 3/6, and \(\mathrm{P}(\mathrm{E} \cap \mathrm{F})\) = 2/6
\[P(E | F)=\frac{P(E \cap F)}{P(F)}=\frac{2 / 6}{3 / 6}=\frac{2}{3} \nonumber.\]
Example \(\PageIndex{5}\)
The following table shows the distribution by gender of students at a community college who take public transportation and the ones who drive to school.
|
Male(M) |
Female(F) |
Non-Binary(N) |
Total |
|
|
Public Transportation(T) |
8 |
10 |
3 |
21 |
|
Drive(D) |
37 |
41 |
1 |
79 |
|
Total |
45 |
51 |
4 |
100 |
The events \(\mathrm{M}, \mathrm{F}, \mathrm{N}, \mathrm{T}\), and \(\mathrm{D}\) are labeled. Find the following probabilities.
- \(\mathrm{P}(\mathrm{D} | \mathrm{N})\)
- \(\mathrm{P}(\mathrm{F} | \mathrm{D})\)
- \(\mathrm{P}(\mathrm{M} | \mathrm{T})\)
Solution 1
Conditional probabilities can often be found directly from a contingency table. If the condition corresponds to only one row or only one column in the table, then you can ignore the rest of the table and read the conditional probability directly from the row or column indicated by the condition.
- The condition is event \(\mathrm{N}\); we can look at only the “Non-Binary” column of the table and ignore the rest of the table: \(\mathrm{P}(\mathrm{D} | \mathrm{N})=\frac{1}{4}\).
- The condition is event \(\mathrm{D}\); we can look at only the “Drive” row of the table and ignore the rest of the table: \(\mathrm{P}(\mathrm{F} | \mathrm{D})=\frac{41}{79}\).
- The condition is event \(\mathrm{T}\); we can look at only the “Public Transportation” row of the table and ignore the rest of the table: \(\mathrm{P}(\mathrm{M} | \mathrm{T})=\frac{8}{21}\).
Solution 2
We use the conditional probability formula \(\mathrm{P}(\mathrm{E} | \mathrm{F})=\frac{\mathrm{P}(\mathrm{E} \cap \mathrm{F})}{\mathrm{P(F)}}\).
- \[P(D | N)=\frac{P(D \cap N)}{P(N)}=\frac{1 / 100}{4 / 100}=\frac{1}{4} \nonumber .\]
- \[P(F | D)=\frac{P(F \cap D)}{P(D)}=\frac{41 / 100}{79 / 100}=\frac{41}{79} \nonumber .\]
- \[\mathrm{P}(\mathrm{M} | \mathrm{T})=\frac{P(M \cap T)}{P(T)}=\frac{8 / 100}{21 / 100}=\frac{8}{21} \nonumber\]
Example \(\PageIndex{6}\)
Given \(\mathrm{P(E)}\) = .5, \(\mathrm{P}(\mathrm{F})\) = .7, and \(\mathrm{P}(\mathrm{E} \cap \mathrm{F})\) = .3. Find the following:
- \(\mathrm{P}(\mathrm{E} | \mathrm{F})\)
- \(\mathrm{P}(\mathrm{F} | \mathrm{E})\)
Solution
We use the conditional probability formula.
- \(P(E | F)=\frac{P(E \cap F)}{P(F)}=\frac{3}{7}=\frac{3}{7}\)
- \(P(F | E)=\frac{P(E \cap F)}{P(E)}=.3 / .5=3 / 5 \)
Example \(\PageIndex{7}\)
\(\mathrm{E}\) and \(\mathrm{F}\) are mutually exclusive events such that \(\mathrm{P(E)}\) = .4, \(\mathrm{P(F)}\) = .9. Find \(\mathrm{P(E | F)}\).
Solution
\(\mathrm{E}\) and \(\mathrm{F}\) are mutually exclusive, so \(\mathrm{P}(\mathrm{E} \cap \mathrm{F})\) = 0.
Therefore \(\mathrm{P}(\mathrm{E} | \mathrm{F})=\frac{\mathrm{P}(\mathrm{E} \cap \mathrm{F})}{\mathrm{P}(\mathrm{F})}=\frac{0}{9}=0\).
Example \(\PageIndex{8}\)
Given \(\mathrm{P}(\mathrm{F} | \mathrm{E})\) = .5, and \(\mathrm{P}(\mathrm{E} \cap \mathrm{F})\) = .3. Find \(\mathrm{P}(\mathrm{E})\).
Solution
Using the conditional probability formula \(\mathrm{P}(\mathrm{E} | \mathrm{F})=\frac{\mathrm{P}(\mathrm{E} \cap \mathrm{F})}{\mathrm{P}(\mathrm{F})}\), we get
\[\mathrm{P}(\mathrm{F} | \mathrm{E})=\frac{\mathrm{P}(\mathrm{E} \cap \mathrm{F})}{\mathrm{P}(\mathrm{E})} \nonumber\]
Substituting and solving:
\[.5=\frac{.3}{\mathrm{P}(\mathrm{E})} \quad \text { or } \quad \mathrm{P}(\mathrm{E})=3 / 5 \nonumber\]
Example \(\PageIndex{9}\)
In a family of three children, find the conditional probability of having two boys and a girl, given that the family has at least two boys.
Solution
Let event \(\mathrm{E}\) be that the family has two boys and a girl, and let \(\mathrm{F}\) be the probability that the family has at least two boys. We want \(\mathrm{P}(\mathrm{E} | \mathrm{F})\).
We list the sample space along with the events \(\mathrm{E}\) and \(\mathrm{F}\).
\begin{aligned}
&\mathrm{S}=\{\mathrm{BBB}, \mathrm{BBG}, \mathrm{BGB}, \mathrm{BGG}, \mathrm{GBB}, \mathrm{GBG}, \mathrm{GGB}, \mathrm{GGG}\}\\
&\mathrm{E}=\{\mathrm{BBG}, \mathrm{BGB}, \mathrm{GBB}\} \text { and } \mathrm{F}=\{\mathrm{BBB}, \mathrm{BBG}, \mathrm{BGB}, \mathrm{GBB}\}\\
&\mathrm{E} \cap \mathrm{F}=\{\mathrm{BBG}, \mathrm{BGB}, \mathrm{GBB}\}
\end{aligned}
Therefore, \(\mathrm{P}(\mathrm{F})\) = 4/8, and \(\mathrm{P}(\mathrm{E} \cap \mathrm{F})\) = 3/8, and
\[\mathrm{P}(\mathrm{E} | \mathrm{F})=\frac{\mathrm{P}(\mathrm{E} \cap \mathrm{F})}{\mathrm{P}(\mathrm{F})} =\frac{3/8}{4/8} =\frac{3}{4} \nonumber.\]
Example \(\PageIndex{10}\)
At a community college 65% of the students subscribe to Amazon Prime, 50% subscribe to Netflix, and 20% subscribe to both. If a student is chosen at random, find the following probabilities:
- the student subscribes to Amazon Prime given that they subscribe to Netflix
- the student subscribes to Netflix given that they subscribe to Amazon Prime
Solution
Let \(\mathrm{A}\) be the event that the student subscribes to Amazon Prime, and \(\mathrm{N}\) be the event that the student subscribes to Netflix.
First identify the probabilities and events given in the problem.
\(\mathrm{P}\)(student subscribes to Amazon Prime) = \(\mathrm{P(A)}\) = 0.65
\(\mathrm{P}\)(student subscribes to Netflix) = \(\mathrm{P(N)}\) = 0.50
\(\mathrm{P}\)(student subscribes to both Amazon Prime and Netflix) = \(\mathrm{P(A \cap N)}\) = 0.20
Then use the conditional probability rule:
- \(\mathrm{P}(\mathrm{A} | \mathrm{N})=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{N})}{\mathrm{P}(\mathrm{N})} = \frac{.20}{.50} = \frac{2}{5}\)
- \(\mathrm{P}(\mathrm{N} | \mathrm{A})=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{N})}{\mathrm{P}(\mathrm{A})} = \frac{.20}{.65} = \frac{4}{13}\)