Quantitative Reasoning

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Question Description

The purpose of this assignment is to use skills you have learned throughout all five weeks of the course. You will be able to apply what they have learned to solve a real world problem that you could use to in your future career.

Create a 5- to 6-slide presentation that must include:

  • One slide on the Introduction
    • Introduce your topic and question that you chose in Week 2.
    • Why did it interest you? How does it relate to life?
    • What should the audience learn from your presentation?
  • Three to four slides of your visuals
    • Show your tables, scatter plot, other 2 visuals, calculations, and any other evidence to support your conclusion(s) that you created in Week 3.
    • Explain what information in the data tables is not needed for your analysis.
    • Discuss what you can conclude from the visuals. How do these visuals support your conclusion?
  • One slide for a conclusion
    • Restate your topic and question and give your answer to the scenario.
    • How confident are you that your conclusion is sound?
    • What work would need to be done to increase your confidence?
    • Discuss what you learned from this project.

Include detailed speaker notes for each slide.

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10/9/2018 University of Phoenix: Using and Understanding Mathematics: A Quantitative Reasoning Approach PRINTED BY: indimac69@email.phoenix.edu. Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. 408 7 PROBABILITY: LIVING WITH THE ODDS Probability is involved in virtually every decision we make. Sometimes the role of probability is clear, as in deciding whether to buy a lottery ticket or whether to plan a picnic based on the probability of rain. In other cases, probability guides decisions on a deeper level. For example, you might choose a particular college because you believe that it is most likely to meet your personal needs. In this chapter, we will look at just how practical and powerful probability can be in our everyday lives. Suppose there are 25 students in your class. What is the probability at least two students in the class have the same birthday (such as February 5 or July 22)? https://phoenix.vitalsource.com/#/books/9781323268674/cfi/6/30!/4/64/4/4/6/2/2/2@0:100 1/157 10/9/2018 University of Phoenix: Using and Understanding Mathematics: A Quantitative Reasoning Approach About 0.01 (or 1 in 100) About 0.25 (or 1 in 4) About 0.6 (or 3 in 5) Exactly 23 365 Exactly 25 365 https://phoenix.vitalsource.com/#/books/9781323268674/cfi/6/30!/4/64/4/4/6/2/2/2@0:100 2/157 10/9/2018 University of Phoenix: Using and Understanding Mathematics: A Quantitative Reasoning Approach PRINTED BY: indimac69@email.phoenix.edu. Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. 408 409 —Cicero (106–43 BC) UNIT 7A Fundamentals of Probability: Explore basic concepts of probability and three methods for determining probabilities: theoretical, relative frequency, and subjective. UNIT 7B Combining Probabilities: Learn basic rules for adding and multiplying probabilities. UNIT 7C The Law of Large Numbers: Explore the law of large numbers and its application to lotteries, insurance, and more. UNIT 7D Assessing Risk: Use probability to describe risk in connection with travel, diseases, and life expectancy. UNIT 7E https://phoenix.vitalsource.com/#/books/9781323268674/cfi/6/30!/4/64/4/4/6/2/2/2@0:100 3/157 10/9/2018 University of Phoenix: Using and Understanding Mathematics: A Quantitative Reasoning Approach Counting and Probability: We study permutations and combinations and apply them to probability and to exploring coincidences. This question may not be particularly “important,” in that knowing the answer is unlikely to affect your daily life. However, it illustrates a very important fact about how our brains think about probability: Research shows that we tend to have poor intuition about probability, and even experts are sometimes wrong about probabilities until they do the detailed calculations. This fact leads many people to underestimate or overestimate probabilities by large margins, which may explain why we often engage in risky activities (such as talking on a cell phone while driving) while fearing low-risk ones (such as flying on a commercial airline). For this reason, the study of probability can be very useful to the decision making we do throughout our lives. In the particular case of the birthday question, most people guess that the probability should be fairly low. After all, there are only 25 students in the class, while there are 365 possible birthdays in a year. Indeed, if we asked the probability that another student in the class had your birthday, it would be close to (but not exactly) 25 , 365 which is less than 0.07, meaning that you’d find someone with your birthday in only about 7 out of 100 randomly selected groups of 25 people. However, the question does not ask about your birthday, but rather about any two students having the same birthday—and this probability turns out to be about 0.57, meaning that there is a better than 50–50 chance of finding two people with the same birthday in any randomly selected group of 25 people. You’ll find the calculation for this surprising answer in Unit 7E, Example 8. https://phoenix.vitalsource.com/#/books/9781323268674/cfi/6/30!/4/64/4/4/6/2/2/2@0:100 4/157 10/9/2018 University of Phoenix: Using and Understanding Mathematics: A Quantitative Reasoning Approach PRINTED BY: indimac69@email.phoenix.edu. Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. 409 410 Lotteries Use this activity to gain a sense of the kinds of problems this chapter will enable you to study. Additional activities are available online in MyMathLab. Lotteries are big business and a major revenue source for many governments. According to the North American Association of State and Provincial Lotteries, all but seven states in the United States have some type of lottery, and in 2010 these lotteries generated a national total of $78 billion in sales. About one-fourth of that money ($19 billion) ended up as state revenue, with the rest going to prizes and expenses. Lotteries present many lessons in probability, and lottery statistics can fuel great debate over whether lotteries are an appropriate way for governments to generate revenue. Working individually or in groups, consider the following questions. Based on the data above and the U.S. population of approximately 315 million, how much does the average American spend on the lottery each year? Surveys indicate that fewer than half of all Americans play the lottery in any given year. What is the average spending per lottery player? Does this result surprise you, or is it consistent with what you would expect based on the spending of your friends or family members who play the lottery? Does your state have a lottery? If so, look up the various games available and the probabilities of winning different prizes. If not, look up the multi-state Powerball game. Consider the fact that lottery operators have spent millions of dollars researching how to encourage people to play the lottery as much as possible. Discuss how this research is applied in the selection of the games and probabilities for your state (or Powerball). Find advertisements for your state lottery (or Powerball). Do the advertisements give a fair assessment of the probabilities you found in question 3? Do they seem deceptive in any way? Explain. https://phoenix.vitalsource.com/#/books/9781323268674/cfi/6/30!/4/64/4/4/6/2/2/2@0:100 5/157 10/9/2018 University of Phoenix: Using and Understanding Mathematics: A Quantitative Reasoning Approach Past studies of gambling suggest that while people in nearly all income groups are almost equally likely to play the lottery, lower-income people tend to play more. Search for statistics concerning the percentage of income spent on the lottery by people in different income groups. Is it fair to say, as critics allege, that lotteries are a form of regressive taxation? Defend your opinion. Based on your answers above and any additional data you can find, discuss the various moral dimensions of lotteries. For example: Are there moral issues related to government sponsorship of gambling? For governments that need more revenue, are lotteries better than higher taxes? Are lotteries unfair to the poor or uneducated? UNIT 7A Fundamentals of Probability Probability plays such a huge role in our lives that it is essential to understand how it works. In this unit, we’ll discuss fundamental concepts required for probability calculations. https://phoenix.vitalsource.com/#/books/9781323268674/cfi/6/30!/4/64/4/4/6/2/2/2@0:100 6/157 10/9/2018 University of Phoenix: Using and Understanding Mathematics: A Quantitative Reasoning Approach PRINTED BY: indimac69@email.phoenix.edu. Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. 410 411 FIGURE 7.1 The four possible outcomes for a toss of two coins. The two middle outcomes both represent the same event of 1 head. Let’s begin by considering a toss of two coins. Figure 7.1 shows that there are four different ways the coins can fall; from left to right in the figure, these are: • Coin 1 and Coin 2 can both land tails. • Coin 1 can land tails and Coin 2 can land heads. • Coin 1 can land heads and Coin 2 can land tails. • Coin 1 and Coin 2 can both land heads. We say that each of these four ways is a different outcome of the coin toss. For convenience, we abbreviate tails with T and heads with H, so we can write the four possible outcomes as TT TH HT HH Note that, when we consider the possible outcomes, the order matters. That is, TH is a different outcome from HT. Now, suppose we are interested only in the number of heads. Because the two middle outcomes (TH and HT) each have 1 head, we say that these two outcomes represent the same event. That is, an event describes one or more possible outcomes that all have the same property of interest—in this case, the same number of heads. Figure 7.1 also shows that if we count the number of heads on two coins, the four possible outcomes represent only three different events: 0 heads (or 0H), 1 head (or 1H), and 2 heads (or 2H). Definitions Outcomes are the most basic possible results of observations or experiments. For example, if you toss two coins, one possible outcome is HT and another possible outcome is TH. An event consists of one or more outcomes that share a property of interest. For example, if you toss two coins and count the number of heads, the outcomes HT and TH both represent https://phoenix.vitalsource.com/#/books/9781323268674/cfi/6/30!/4/64/4/4/6/2/2/2@0:100 7/157 10/9/2018 University of Phoenix: Using and Understanding Mathematics: A Quantitative Reasoning Approach the same event of 1 head (and 1 tail). EXAMPLE 1 Family Outcomes and Events Consider families of two children. List all the possible outcomes for the birth order of boys and girls. If we are only interested in the total number of boys in the families, what are the possible events? Solution Using B to represent a boy and G to represent a girl, there are four different possible birth orders (outcomes): BB, BG, GB, and GG. Because we are asked to consider only the total number of boys, the possible events with two children are: 0 boys, 1 boy, and 2 boys. The event 0 boys (0B) consists of the single outcome GG, the event 1 boy (1B) consists of the outcomes GB and BG, and the event 2 boys (2B) consists of the single outcome BB. Now try Exercises 17–18. Finding Probabilities Let’s think about a single coin and assume that it is “fair,” meaning that it is equally likely to land heads (H) or tails (T). In everyday language, we say that the chance of the coin landing heads on a single toss is “50–50,” which means that we expect the coin to https://phoenix.vitalsource.com/#/books/9781323268674/cfi/6/30!/4/64/4/4/6/2/2/2@0:100 8/157 10/9/2018 University of Phoenix: Using and Understanding Mathematics: A Quantitative Reasoning Approach PRINTED BY: indimac69@email.phoenix.edu. Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. land heads 50% of the time and tails 50% of the time. However, for the purposes of 411 412 calculation, it is better to express the probability as a fraction. With one toss of the coin, there are two equally likely outcomes: H and T. If the event of interest is the coin landing heads (H), then the probability of this event is 1 out of 2, or 1/2, because H is 1 of the 2 equally likely outcomes (H and T). Using the letter P to represent probability, we express the probability of heads as follows: P (H) = 1 2 = 0.5 We read this statement as “the probability of heads equals one-half, or 0.5.” More generally, we use the notation P(event) to mean the probability of any event; we often denote events by letters or symbols, as we did in using H for heads. We can use the same notation when the event of interest for the single coin is that it lands tails, which also has a probability of 1/2: P (T ) = 1 2 = 0.5 Notice that, because no outcomes besides heads (H) and tails (T) are possible, it is certain that we will toss either a head or tail. That is, the probability of an event in which the coin lands as either a head or tail is 2 out of 2, or 2/2 = 1: P (coin lands either heads or tails) = 2 2 =1 BY THE WAY In very rare cases, a coin can land on its edge. However, left to itself, the coin will eventually fall over, which is why heads and tails are considered the only possible outcomes. Still, you might think of exceptions under special circumstances. For example, if you toss a coin while in the International Space Station, it will float weightlessly along with everything else. https://phoenix.vitalsource.com/#/books/9781323268674/cfi/6/30!/4/64/4/4/6/2/2/2@0:100 9/157 10/9/2018 University of Phoenix: Using and Understanding Mathematics: A Quantitative Reasoning Approach In other words, an event that is certain to occur has a probability of 1. Similarly, because H and T are the only possible outcomes, any event that does not include either of these outcomes has a probability of zero. For example, there is no chance that the coin will “remain balanced on its edge forever,” so P(coin remains balanced on its edge forever) = 0. Generalizing the coin toss example, probabilities always have values between 0 and 1. A probability of 0 means an event is impossible, and a probability of 1 is certain. In between, larger fractions mean larger probabilities. Figure 7.2 shows the scale of probability values, along with common expressions of likelihood. FIGURE 7.2 The scale shows various degrees of certainty as expressed by probabilities. Expressing Probability The probability of an event, expressed as P (event), is always between 0 and 1 (inclusive). A probability of 0 means the event is impossible and a probability of 1 means the event is certain. Now that you understand how to express probabilities, we can consider how to calculate or estimate them. There are three basic techniques for finding probabilities, which we call the theoretical method, the relative frequency method, and the subjective method. We’ll examine each in turn. Time Out to Think https://phoenix.vitalsource.com/#/books/9781323268674/cfi/6/30!/4/64/4/4/6/2/2/2@0:100 10/157 10/9/2018 University of Phoenix: Using and Understanding Mathematics: A Quantitative Reasoning Approach Place each of the following events on the scale in Figure 7.2, and explain how you chose where to put them: (a) the event of the Sun being above the horizon in the daytime; (b) the event of being in two places at the same time; (c) the event of being hit by a bus; (d) the event of getting an A in your math class. https://phoenix.vitalsource.com/#/books/9781323268674/cfi/6/30!/4/64/4/4/6/2/2/2@0:100 11/157 10/9/2018 University of Phoenix: Using and Understanding Mathematics: A Quantitative Reasoning Approach PRINTED BY: indimac69@email.phoenix.edu. Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. 412 413 Theoretical Probabilities When we say that the probability of heads on a coin toss is 1/2, we are assuming that the coin is fair and is equally likely to land heads or tails. In essence, the probability is based on a theory of how the coin behaves, so we say that the probability of 1/2 comes from the theoretical method. More generally, we can apply the theoretical method to find probabilities in any situation for which we know all the possible outcomes and all the outcomes are equally likely. The following box summarizes the procedure. BY THE WAY Theoretical methods are also called a priori methods. The words a priori are Latin for “before the fact” or “before experience.” Theoretical Method for Equally Likely Outcomes Step 1. Count the total number of possible outcomes. Step 2. Among all the possible outcomes, count the number of ways the event of interest, A, can occur. Step 3. Determine the probability, P(A), from p (A) = numberofwaysAcanoccur totalnumberofpossibleoutcomes To see how the procedure works, look back at Figure 7.1, which shows the four possible outcomes for a two-coin toss (HH, HT, TH, and TT). We can apply the theoretical method because these are the only four possible outcomes and they are all equally likely. If we are interested in the event of 2 heads, Figure 7.1 shows that there is only one way (HH) that it can occur. Therefore, its probability is p (2 heads) = number of ways 2 heads can occur total number of possible outcomes = 1 4 EXAMPLE 2 Coins and Dice https://phoenix.vitalsource.com/#/books/9781323268674/cfi/6/30!/4/64/4/4/6/2/2/2@0:100 12/157 10/9/2018 University of Phoenix: Using and Understanding Mathematics: A Quantitative Reasoning Approach Apply the theoretical method to find the probability of: a. exactly one head when you toss two coins b. getting a 3 when you roll a 6-sided die Solution a. The four possible and equally likely outcomes for tossing two coins are shown in Figure 7.1. Two of these four outcomes represent the event of exactly one head (TH and HT). Therefore, the probability of exactly one head is p (1 head) = number of ways 1 heads can occur total number of possible outcomes = 2 4 = 1 2 When tossing two coins together, the probability of getting exactly one head is 1/2, meaning that we expect to get one head (and one tail) about half of the time. b. Figure 7.3 shows the six possible outcomes for rolling a single 6-sided die. The six outcomes are all equally likely and only one is an outcome of “3,” so the probability of rolling a 3 is p (rolling a3) = number of ways “3” can occur total number of possible outcomes = 1 6 FIGURE 7.3 The six possible outcomes for a roll of one die. TECHNICAL NOTE Except when stated otherwise, we assume that dice are fair, six-sided cubes with face values of 1 through 6. https://phoenix.vitalsource.com/#/books/9781323268674/cfi/6/30!/4/64/4/4/6/2/2/2@0:100 13/157 10/9/2018 University of Phoenix: Using and Understanding Mathematics: A Quantitative Reasoning Approach PRINTED BY: indimac69@email.phoenix.edu. Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. 413 414 The probability of rolling a 3 on a six-sided die is 1/6; that is, if you roll the die many times, you should see a 3 about 1/6 of the time. Now try Exercises 19–20. EXAMPLE 3 Playing Card Probabilities Figure 7.4 shows the 52 playing cards in a standard deck. There are four suits, known as hearts, spades, diamonds, and clubs. Each suit has cards for the numbers 2 through 10 plus a jack, queen, king, and ace (for a total of 13 cards in each suit). Notice that hearts and diamonds are red, while spades and clubs are black. If you draw one card at random from a standard deck, what is the probability that it is a spade? FIGURE 7.4 Playing cards in a standard 52-card deck. Solution Let’s proceed step by step with the theoretical method, which applies because each of the 52 cards is equally likely to be drawn. Step 1. Each card represents a possible outcome, so there are 52 possible outcomes. Step 2. The event of interest is drawing a spade, and there are 13 spades in the deck. Step 3. The probability that a randomly drawn card is a spade is p (spade) = number of outcomes that are spades total number of possible outcomes = 13 52 = 1 4 Now tr ...
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NicholasI
School: UIUC

Attached.

Security and Criminal Justice Scenario
Student Name
Institution
Date

Introduction


Topic: Security and Criminal Justice scenario.



Question: What is the number of the United States population that will be supervised by U.S. adult
correctional system in 2018.



I chose this topic because I wanted to understand the justice system and how it works. I also wanted to
know the total number of people in prison so that I can figure out how serious the issue of security is
through the analysis of the correctional population over the years.



In predicting the correctional population under supervision in...

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