Using and Understanding Mathematics: A Quantitative Reasoning Approach (6th Edition)
6th Edition
ISBN: 9780321914620
Author: Jeffrey O. Bennett, William L. Briggs
Publisher: PEARSON
expand_more
expand_more
format_list_bulleted
Concept explainers
Videos
Textbook Question
Chapter 7.B, Problem 6QQ
You roll two dice. Based on the probabilities shown in Table 7.3, which of the following has a probability greater than 0.5?
- rolling a sum of 2 or 3 or 4 or 5
- rolling a sum of 2 or 3 or 4 or 10 or 11 or 12
- rolling a sum of 5 or 6 or 7 or 8
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
5.
Let a probability measure P on ([0,3], B([0,3])) be given by
1
dP(s):
=
½ s² ds.
9
Consider a random variable X : [0,3]
→ R given by
X(s) = s², sc [0,3].
S
Total marks 7
Find the distribution of X.
[7 Marks]
Refer to page 24 for solving a differential equation using Laplace transforms.
Instructions:
Take the Laplace transform of the given equation, applying initial conditions appropriately.
⚫ Solve the resulting algebraic equation and find the inverse transform.
Provide step-by-step solutions with intermediate results and final verification.
Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]
Refer to page 30 for deriving the Euler-Lagrange equation for an optimal control problem.
Instructions:
•
Use the calculus of variations to derive the Euler-Lagrange equation.
Clearly define the functional being minimized or maximized.
Provide step-by-step derivations, including all necessary boundary conditions. Avoid skipping
critical explanations.
Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440 AZF/view?usp=sharing]
Chapter 7 Solutions
Using and Understanding Mathematics: A Quantitative Reasoning Approach (6th Edition)
Ch. 7.A - Suppose you toss one coin three times in a row and...Ch. 7.A - During the course of the basketball season, Shawna...Ch. 7.A - A box contains 20 chocolates, but only 2 of them...Ch. 7.A - Prob. 4QQCh. 7.A - When you toss one coin, the probability that...Ch. 7.A - On a roll of two dice, Serena bets that the sum...Ch. 7.A - Suppose you toss four 6-sided dice. How many...Ch. 7.A - Suppose you toss four 6-sided dice. How many...Ch. 7.A - You are playing 5-card poker with a deck of 52...Ch. 7.A - Prob. 10QQ
Ch. 7.A - Distinguish between an outcome and an event in...Ch. 7.A - 2. What does it mean when we write P(event)? What...Ch. 7.A - Briefly describe the differences among...Ch. 7.A - How is the probability of an event not occurring...Ch. 7.A - Prob. 5ECh. 7.A - Prob. 6ECh. 7.A - When I toss four coins, there are four different...Ch. 7.A - The probability that my sister will get into the...Ch. 7.A - I estimate that the probability of my getting...Ch. 7.A - Because either there is life on Mars or there is...Ch. 7.A - The probability that Jonas will win the race is...Ch. 7.A - Based on data showing that we’ve had snow on...Ch. 7.A - Prob. 13ECh. 7.A - 13-16: Review of the Multiplication Principle. Use...Ch. 7.A - Review of the Multiplication Principle. Use the...Ch. 7.A - Prob. 16ECh. 7.A - Prob. 17ECh. 7.A - Double-Header Outcomes and Events. Suppose the New...Ch. 7.A - Prob. 19ECh. 7.A - Prob. 20ECh. 7.A - Prob. 21ECh. 7.A - Prob. 22ECh. 7.A - Prob. 23ECh. 7.A - Prob. 24ECh. 7.A - 19-28: Theoretical Probabilities. Use the...Ch. 7.A - Theoretical Probabilities. Use the theoretical...Ch. 7.A - 19-28: Theoretical Probabilities. Use the...Ch. 7.A - 19-28: Theoretical Probabilities. Use the...Ch. 7.A - Prob. 29ECh. 7.A - Prob. 30ECh. 7.A - Prob. 31ECh. 7.A - Prob. 32ECh. 7.A - Which Type of Probability? State which method...Ch. 7.A - 33-34: Which Type of Probability? State which...Ch. 7.A - Prob. 35ECh. 7.A - Prob. 36ECh. 7.A - Event Not Occurring. Determine the probability of...Ch. 7.A - Prob. 38ECh. 7.A - Probability Distributions. Make a probability...Ch. 7.A - 39-42: Probability Distributions. Make a...Ch. 7.A - 39-42: Probability Distributions. Make a...Ch. 7.A - 39-42: Probability Distributions. Make a...Ch. 7.A - Prob. 43ECh. 7.A - Prob. 44ECh. 7.A - Prob. 45ECh. 7.A - Prob. 46ECh. 7.A - Prob. 47ECh. 7.A - Prob. 48ECh. 7.A - Prob. 49ECh. 7.A - Prob. 50ECh. 7.A - Prob. 51ECh. 7.A - Prob. 52ECh. 7.A - Prob. 53ECh. 7.A - Prob. 54ECh. 7.A - Computing Probabilities. Decide which method...Ch. 7.A - Computing Probabilities. Decide which method...Ch. 7.A - Prob. 57ECh. 7.A - Prob. 58ECh. 7.A - Prob. 59ECh. 7.A - Computing Probabilities. Decide which method...Ch. 7.A - Prob. 61ECh. 7.A - Prob. 62ECh. 7.A - Prob. 63ECh. 7.A - Prob. 64ECh. 7.A - Prob. 65ECh. 7.A - Prob. 66ECh. 7.A - Prob. 67ECh. 7.A - Probability Distributions 68.Suppose you have a...Ch. 7.A - Prob. 69ECh. 7.A - More Counting. Answer the following counting...Ch. 7.A - Prob. 71ECh. 7.A - Prob. 72ECh. 7.A - Prob. 73ECh. 7.A - Prob. 74ECh. 7.A - Senior Citizens. In 2010, there were 40 million...Ch. 7.A - Prob. 76ECh. 7.A - Prob. 77ECh. 7.A - Prob. 78ECh. 7.A - Prob. 79ECh. 7.A - Prob. 80ECh. 7.A - Prob. 81ECh. 7.A - Probability in the News. Find a news article or...Ch. 7.A - Prob. 83ECh. 7.A - Gambling Odds. Find an advertising for a gambling...Ch. 7.B - The probability of rolling two dice and getting a...Ch. 7.B - The rule PAandB=PAPBholdsa. in all cases.b. only...Ch. 7.B - Prob. 3QQCh. 7.B - A box of candy contains five dark chocolates and...Ch. 7.B - Prob. 5QQCh. 7.B - You roll two dice. Based on the probabilities...Ch. 7.B - You roll two dice twice. Based on the...Ch. 7.B - You toss two coins ten times, and you want to know...Ch. 7.B - Prob. 9QQCh. 7.B - Prob. 10QQCh. 7.B - How did the gambling habits of the Chevalier de...Ch. 7.B - 2. Give an example in which we would be interested...Ch. 7.B - 3. Give an example in which we would be interested...Ch. 7.B - 4. What is the at least once rule? Explain how the...Ch. 7.B - The probability of getting heads and tails when...Ch. 7.B - If you toss a corn and get heads three times in a...Ch. 7.B - The probability of drawing ace or a spade from a...Ch. 7.B - I can’t believe you chose the lottery number...Ch. 7.B - Prob. 9ECh. 7.B - To find the probability that at least one of my 25...Ch. 7.B - Chevalier’s Logic with Coins. The Chevalier’s...Ch. 7.B - Prob. 12ECh. 7.B - Prob. 13ECh. 7.B - Prob. 14ECh. 7.B - Prob. 15ECh. 7.B - Prob. 16ECh. 7.B - Prob. 17ECh. 7.B - Prob. 18ECh. 7.B - Prob. 19ECh. 7.B - Prob. 20ECh. 7.B - Prob. 21ECh. 7.B - Prob. 22ECh. 7.B - Prob. 23ECh. 7.B - 2 1-24: Either/Or Probabilities. Determine whether...Ch. 7.B - Prob. 25ECh. 7.B - 26. Randomly meeting three international students...Ch. 7.B - Prob. 27ECh. 7.B - Prob. 28ECh. 7.B - Prob. 29ECh. 7.B - Prob. 30ECh. 7.B - At Least Once Problems. Use the at least once rule...Ch. 7.B - Prob. 32ECh. 7.B - Prob. 33ECh. 7.B - Prob. 34ECh. 7.B - Prob. 35ECh. 7.B - Prob. 36ECh. 7.B - Prob. 37ECh. 7.B - Prob. 38ECh. 7.B - Prob. 39ECh. 7.B - Prob. 40ECh. 7.B - Prob. 41ECh. 7.B - Prob. 42ECh. 7.B - Prob. 43ECh. 7.B - Prob. 44ECh. 7.B - Prob. 45ECh. 7.B - Prob. 46ECh. 7.B - Assorted Probabilities. Use the method of your...Ch. 7.B - Assorted Probabilities. Use the method of your...Ch. 7.B - Prob. 49ECh. 7.B - Prob. 50ECh. 7.B - Prob. 51ECh. 7.B - Prob. 52ECh. 7.B - Assorted Probabilities. Use the method of your...Ch. 7.B - Prob. 54ECh. 7.B - Assorted Probabilities. Use the method of your...Ch. 7.B - Assorted Probabilities. Use the method of your...Ch. 7.B - Probability and Court. The data in the following...Ch. 7.B - Testing a Drug. A new cold medication was tested...Ch. 7.B - Prob. 59ECh. 7.B - Dominant and Recessive Genes. Many traits are...Ch. 7.B - Better Bet for the Chevalier. Suppose that the...Ch. 7.B - Prob. 62ECh. 7.B - Miami Hurricanes. Studies of the Florida...Ch. 7.B - Prob. 64ECh. 7.B - Prob. 65ECh. 7.B - Prob. 66ECh. 7.B - Prob. 67ECh. 7.B - Combined Probability in Your Life. Cite a recent...Ch. 7.C - Suppose that the probability of a hurricane...Ch. 7.C - Consider a lottery with 100 million tickets in...Ch. 7.C - Consider the lottery described in Exercise 2. If...Ch. 7.C - Prob. 4QQCh. 7.C - An insurance company knows that the average cost...Ch. 7.C - You know a shortcut to work that uses side streets...Ch. 7.C - Cameron is betting on a game in which the...Ch. 7.C - Prob. 8QQCh. 7.C - A $1 slot at a casino is set so that it returns...Ch. 7.C - Consider the slot machine described in Exercise 9....Ch. 7.C - Explain the meaning of the law of large numbers....Ch. 7.C - In 10 tosses of a fair coin, should you be...Ch. 7.C - Prob. 3ECh. 7.C - Prob. 4ECh. 7.C - Prob. 5ECh. 7.C - Prob. 6ECh. 7.C - The expected value to me of each raffle ticket I...Ch. 7.C - The expected value of each insurance policy our...Ch. 7.C - If you toss a coin four times, it’s much more...Ch. 7.C - I haven’t won in my last 25 pulls on the slot...Ch. 7.C - I haven’t won in my last 25 pulls on the slot...Ch. 7.C - I’ve lost S750 so far today on roulette. I’m going...Ch. 7.C - Understanding the Law of Large Numbers. Suppose...Ch. 7.C - Speedy Driver. Suppose a person who has a habit of...Ch. 7.C - Prob. 15ECh. 7.C - Expected Value Games. Find the expected value (to...Ch. 7.C - Prob. 17ECh. 7.C - Expected Value Games. Find the expected value (to...Ch. 7.C - Prob. 19ECh. 7.C - Prob. 20ECh. 7.C - Prob. 21ECh. 7.C - Expected Wait. A bus arrives at a bus stop at...Ch. 7.C - Gambler’s Fallacy and Coins. Suppose you play a...Ch. 7.C - Gambler’s Fallacy and Dice. Suppose you roll a die...Ch. 7.C - Can You Catch Up? Suppose you toss a fair coin 100...Ch. 7.C - Prob. 26ECh. 7.C - Prob. 27ECh. 7.C - Prob. 28ECh. 7.C - Prob. 29ECh. 7.C - Prob. 30ECh. 7.C - Prob. 31ECh. 7.C - Prob. 32ECh. 7.C - Prob. 33ECh. 7.C - Prob. 34ECh. 7.C - Extra Points in Football. Football teams have the...Ch. 7.C - Roulette. When you bet $5 on the number 7 in...Ch. 7.C - Household Size. It is estimated that 57% of...Ch. 7.C - Prob. 38ECh. 7.C - Prob. 39ECh. 7.C - Prob. 40ECh. 7.C - Prob. 41ECh. 7.C - The Morality of Gambling. Republican Senator...Ch. 7.D - Prob. 1QQCh. 7.D - Prob. 2QQCh. 7.D - Prob. 3QQCh. 7.D - Prob. 4QQCh. 7.D - Prob. 5QQCh. 7.D - Based on Figure 7.11, besides the elderly, the...Ch. 7.D - Prob. 7QQCh. 7.D - Prob. 8QQCh. 7.D - Based on Figure 7.12, which statement is not...Ch. 7.D - Prob. 10QQCh. 7.D - Prob. 1ECh. 7.D - Prob. 2ECh. 7.D - Prob. 3ECh. 7.D - Prob. 4ECh. 7.D - Prob. 5ECh. 7.D - Prob. 6ECh. 7.D - Your life expectancy is the major factor in...Ch. 7.D - A 60-year-old has a shorter life expectancy than...Ch. 7.D - Prob. 9ECh. 7.D - Prob. 10ECh. 7.D - Prob. 11ECh. 7.D - Prob. 12ECh. 7.D - Prob. 13ECh. 7.D - Prob. 14ECh. 7.D - Prob. 15ECh. 7.D - 15–20: Causes of Death. Use Table 7.4, and assume...Ch. 7.D - Prob. 17ECh. 7.D - Prob. 18ECh. 7.D - 15–20: Causes of Death. Use Table 7.4, and assume...Ch. 7.D - 15–20: Causes of Death. Use Table 7.4, and assume...Ch. 7.D - Prob. 21ECh. 7.D - Prob. 22ECh. 7.D - Prob. 23ECh. 7.D - Prob. 24ECh. 7.D - Prob. 25ECh. 7.D - Prob. 26ECh. 7.D - Prob. 27ECh. 7.D - Prob. 28ECh. 7.D - Prob. 29ECh. 7.D - Prob. 30ECh. 7.D - Prob. 31ECh. 7.D - Prob. 32ECh. 7.D - Prob. 33ECh. 7.D - Aging population. The table shows the U S....Ch. 7.D - Prob. 35ECh. 7.D - Prob. 36ECh. 7.D - Prob. 37ECh. 7.D - Prob. 38ECh. 7.D - Prob. 39ECh. 7.E - 1. You are asked to create a 4-character password,...Ch. 7.E - Prob. 2QQCh. 7.E - A teacher has 28 students. and 5 them will be...Ch. 7.E - Prob. 4QQCh. 7.E - A soccer coach who has 15 children on her team...Ch. 7.E - One term in the denominator of the combinations...Ch. 7.E - Overall, number of different 4-person terns (order...Ch. 7.E - One in a stadium filled with 100,000 people is...Ch. 7.E - One in a stadium filled With people is chosen at...Ch. 7.E - There are 365 possible birthdays in a year. In a...Ch. 7.E - Prob. 1ECh. 7.E - Prob. 2ECh. 7.E - Prob. 3ECh. 7.E - 4. Explain What we mean When we that outcome is...Ch. 7.E - I used the permutations formula to determine how...Ch. 7.E - I used the combinations formula to determine how...Ch. 7.E - The number Of different possible batting orders...Ch. 7.E - It must my lucky day, the five-card poker hand I...Ch. 7.E - The probability that two people in a randomly...Ch. 7.E - Someone wins the lottery every week. so I figure...Ch. 7.E - Review of Factorials. use the skills covered in...Ch. 7.E - Review of Factorials. use the skills covered in...Ch. 7.E - Review of Factorials. use the skills covered in...Ch. 7.E - Prob. 14ECh. 7.E - Prob. 15ECh. 7.E - Prob. 16ECh. 7.E - Prob. 17ECh. 7.E - Prob. 18ECh. 7.E - Review off Factorials. use the skills covered in...Ch. 7.E - 11—22: Review off Factorials. use the skills...Ch. 7.E - 11—22: Review off Factorials. use the skills...Ch. 7.E - 11—22: Review off Factorials. use the skills...Ch. 7.E - Prob. 23ECh. 7.E - Counting Methods. Answer the following questions...Ch. 7.E - Prob. 25ECh. 7.E - Prob. 26ECh. 7.E - Prob. 27ECh. 7.E - Prob. 28ECh. 7.E - Prob. 29ECh. 7.E - Prob. 30ECh. 7.E - Prob. 31ECh. 7.E - Prob. 32ECh. 7.E - Counting Methods. Answer the following questions...Ch. 7.E - Prob. 34ECh. 7.E - Prob. 35ECh. 7.E - Prob. 36ECh. 7.E - Prob. 37ECh. 7.E - Prob. 38ECh. 7.E - Prob. 39ECh. 7.E - Prob. 40ECh. 7.E - Birthday Coincidences. Suppose you are part of a...Ch. 7.E - Birthday Coincidences. Suppose you are part of a...Ch. 7.E - 43. Ice Cream Shop. Josh and Ice Cream Shop Of-...Ch. 7.E - 44. Telephone Numbers. A ten-digit phone number in...Ch. 7.E - 45. Pizza Hype. Luigi; Pizza parlor advertises 56...Ch. 7.E - ZIP Codes. The US. Postal Service five-digit and...Ch. 7.E - 47 —54: Counting and Probability. Find the...Ch. 7.E - Prob. 48ECh. 7.E - Prob. 49ECh. 7.E - Prob. 50ECh. 7.E - Prob. 51ECh. 7.E - Prob. 52ECh. 7.E - Prob. 53ECh. 7.E - Prob. 54ECh. 7.E - Hot Streaks. Suppose that 2000 people are all...Ch. 7.E - Joe Record. One of longest-standing records in...Ch. 7.E - Prob. 57ECh. 7.E - Prob. 58ECh. 7.E - Prob. 59ECh. 7.E - Prob. 60ECh. 7.E - Prob. 61ECh. 7.E - Prob. 62ECh. 7.E - Prob. 63ECh. 7.E - Prob. 64E
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.Similar questions
- Refer to page 32 for solving a linear-quadratic regulator (LQR) problem. Instructions: • Formulate the cost functional and state-space representation. • Derive the Riccati equation and solve it step-by-step. Clearly explain how the optimal control law is obtained. Ensure all matrix algebra is shown in detail. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]arrow_forwardRefer to page 14 for solving a linear first-order differential equation. Instructions: • Convert the equation into its standard linear form. • Use integrating factors to find the solution. Show all steps explicitly, from finding the factor to integrating and simplifying the solution. Link [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]arrow_forwardRefer to page 10 for a problem involving solving an exact differential equation. Instructions: • Verify if the equation is exact by testing әм მყ - ƏN მე If not exact, determine an integrating factor to make it exact. • Solve step-by-step, showing all derivations. Avoid irrelevant explanations. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Haz b9tC440AZF/view?usp=sharing]arrow_forward
- Refer to page 10 for a problem involving solving an exact differential equation. Instructions: Verify exactness carefully. ⚫ If the equation is not exact, find an integrating factor to make it exact. Solve step-by-step and ensure no algebraic steps are skipped. Provide detailed explanations for each transformation. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 34 for deriving and applying Pontryagin's Maximum Principle. Instructions: ⚫ Define the Hamiltonian for the given control problem. • • Derive the necessary conditions for optimality step-by-step, including state and co-state equations. Solve the resulting system of equations explicitly, showing all intermediate steps. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]arrow_forwardRefer to page 20 for solving a separable differential equation. Instructions: ⚫ Separate the variables explicitly. • Integrate both sides carefully, showing intermediate steps. • Simplify the final result and provide the explicit or implicit solution as required. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]arrow_forward
- Refer to page 16 for a problem involving solving a second-order linear homogeneous differential equation. Instructions: • Analyze the characteristic equation and address all possible cases (distinct, repeated, and complex roots). • Show detailed steps for deriving the general solution. • Verify solutions by substitution into the original equation. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]arrow_forwardNeed help with question?arrow_forwardNeed help with question?arrow_forward
- Refer to page 15 for a problem involving evaluating a double integral in polar coordinates. Instructions: Convert the given Cartesian integral to polar coordinates. Show all transformations and step-by-step calculations. Link [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]arrow_forwardRefer to page 9 for a problem requiring finding the tangent plane to a given surface at a point. Instructions: Use partial derivatives to calculate the equation of the tangent plane. Show all calculations step-by-step. Link [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]arrow_forwardRefer to page 8 for a problem involving solving a second-order linear homogeneous differential equation. Instructions: Solve using characteristic equations. Show all intermediate steps leading to the general solution. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440AZF/view?usp=sharing]arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Holt Mcdougal Larson Pre-algebra: Student Edition...AlgebraISBN:9780547587776Author:HOLT MCDOUGALPublisher:HOLT MCDOUGAL
Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning
Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
College AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning
College Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage Learning
Holt Mcdougal Larson Pre-algebra: Student Edition...
Algebra
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
College Algebra
Algebra
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:Cengage Learning
College Algebra (MindTap Course List)
Algebra
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:Cengage Learning
Bayes' Theorem 1: Introduction and conditional probability; Author: Dr Nic's Maths and Stats;https://www.youtube.com/watch?v=lQVkXfJ-rpU;License: Standard YouTube License, CC-BY
What is Conditional Probability | Bayes Theorem | Conditional Probability Examples & Problems; Author: ACADGILD;https://www.youtube.com/watch?v=MxOny_1y2Q4;License: Standard YouTube License, CC-BY
Bayes' Theorem of Probability With Tree Diagrams & Venn Diagrams; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=OByl4RJxnKA;License: Standard YouTube License, CC-BY
Bayes' Theorem - The Simplest Case; Author: Dr. Trefor Bazett;https://www.youtube.com/watch?v=XQoLVl31ZfQ;License: Standard Youtube License