Using & Understanding Mathematics: A Quantitative Reasoning Approach (7th Edition)
7th Edition
ISBN: 9780134705187
Author: Jeffrey O. Bennett, William L. Briggs
Publisher: PEARSON
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Chapter 7.B, Problem 16E
And Probabilities. Determine whether the events described in each exercise are independent or dependent. Then find the and probability of the events.
16. The next five births at a hospital all being boys
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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 & Understanding Mathematics: A Quantitative Reasoning Approach (7th 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 - Suppose the probability of winning a certain prize...Ch. 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 - The odds on (or odds against) TripleTreat winning...
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 - Review of the Multiplication Principle. Use the...Ch. 7.A - Double-Header Outcomes and Events. Suppose the New...Ch. 7.A - Prob. 18ECh. 7.A - Theoretical Probabilities. Use the theoretical...Ch. 7.A - Theoretical Probabilities. Use the theoretical...Ch. 7.A - Theoretical Probabilities. Use the theoretical...Ch. 7.A - Theoretical Probabilities. Use the theoretical...Ch. 7.A - Theoretical Probabilities. Use the theoretical...Ch. 7.A - Theoretical Probabilities. Use the theoretical...Ch. 7.A - Theoretical Probabilities. Use the theoretical...Ch. 7.A - Theoretical Probabilities. Use the theoretical...Ch. 7.A - 19-28: Theoretical Probabilities. Use the...Ch. 7.A - Theoretical Probabilities. Use the theoretical...Ch. 7.A - Relative Frequency Probabilities. Use the relative...Ch. 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 - Event Not Occurring. Determine the probability of...Ch. 7.A - Event Not Occurring. Determine the probability of...Ch. 7.A - Event Not Occurring. Determine the probability of...Ch. 7.A - Event Not Occurring. Determine the probability of...Ch. 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 - Odds. Use the definition given in the text to find...Ch. 7.A - 43-46: Odds. Use the definition given in the text...Ch. 7.A - 43-46: Odds. Use the definition given in the text...Ch. 7.A - 43-46: Odds. Use the definition given in the text...Ch. 7.A - Gambling Odds. Use the definition of odds in...Ch. 7.A - Prob. 48ECh. 7.A - 49-66: Computing Probabilities. Decide which...Ch. 7.A - 49-66: Computing Probabilities. Decide which...Ch. 7.A - Computing Probabilities. Decide which method...Ch. 7.A - Computing Probabilities. Decide which method...Ch. 7.A - Prob. 53ECh. 7.A - Prob. 54ECh. 7.A - Prob. 55ECh. 7.A - Computing Probabilities. Decide which method...Ch. 7.A - Prob. 57ECh. 7.A - Computing Probabilities. Decide which method...Ch. 7.A - Prob. 59ECh. 7.A - Prob. 60ECh. 7.A - Computing Probabilities. Decide which method...Ch. 7.A - 49-66: Computing Probabilities. Decide which...Ch. 7.A - 49-66: Computing Probabilities. Decide which...Ch. 7.A - 49-66: Computing Probabilities. Decide which...Ch. 7.A - Computing Probabilities. Decide which method...Ch. 7.A - Computing Probabilities. Decide which method...Ch. 7.A - Probability Distributions 68.Suppose you have a...Ch. 7.A - Prob. 68ECh. 7.A - More Counting. Answer the following counting...Ch. 7.A - Prob. 70ECh. 7.A - Prob. 71ECh. 7.A - Prob. 72ECh. 7.A - 73. Gender Politics. The following table gives...Ch. 7.A - 74. Senior Citizens. ln 2015, there were 48...Ch. 7.A - Marriage Status. The following table gives...Ch. 7.A - Prob. 76ECh. 7.A - Prob. 77ECh. 7.A - Prob. 78ECh. 7.A - Prob. 79ECh. 7.A - Prob. 80ECh. 7.A - Probability in the News. Find a news article or...Ch. 7.A - Prob. 82ECh. 7.A - Gambling Odds. Find an advertising for a gambling...Ch. 7.A - Prob. 84ECh. 7.A - 85. Dice Simulation. Follow the method used in...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 - In which of the following cases are the events...Ch. 7.B - A box of candy contains five dark chocolates and...Ch. 7.B - The events of being born on a Monday and being...Ch. 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 - You purchase 10 lottery tickets for which the...Ch. 7.B - One in 10 people on campus has blond hair. In 20...Ch. 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 - My chance of getting a 5 on a roll of one die is...Ch. 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 - And Probabilities. Determine whether the events...Ch. 7.B - And Probabilities. Determine whether the events...Ch. 7.B - And Probabilities. Determine whether the events...Ch. 7.B - And Probabilities. Determine whether the events...Ch. 7.B - And Probabilities. Determine whether the events...Ch. 7.B - And Probabilities. Determine whether the events...Ch. 7.B - And Probabilities. Determine whether the events...Ch. 7.B - And Probabilities. Determine whether the events...Ch. 7.B - And Probabilities. Determine whether the events...Ch. 7.B - And Probabilities. Determine whether the events...Ch. 7.B - Either/Or Probabilities. Determine whether the...Ch. 7.B - Either/Or Probabilities. Determine whether the...Ch. 7.B - Either/Or Probabilities. Determine whether the...Ch. 7.B - Either/Or Probabilities. Determine whether the...Ch. 7.B - Either/Or Probabilities. Determine whether the...Ch. 7.B - Either/Or Probabilities. Determine whether the...Ch. 7.B - Prob. 29ECh. 7.B - At Least Once Problems. Use the at least once rule...Ch. 7.B - At Least Once Problems. Use the at least once rule...Ch. 7.B - At Least Once Problems. Use the at least once rule...Ch. 7.B - At Least Once Problems. Use the at least once rule...Ch. 7.B - At Least Once Problems. Use the at least once rule...Ch. 7.B - Prob. 35ECh. 7.B - Assorted Probabilities. Use the method of your...Ch. 7.B - Prob. 37ECh. 7.B - Assorted Probabilities. Use the method of your...Ch. 7.B - Assorted Probabilities. Use the method of your...Ch. 7.B - Prob. 40ECh. 7.B - Prob. 41ECh. 7.B - Prob. 42ECh. 7.B - Assorted Probabilities. Use the method of your...Ch. 7.B - Assorted Probabilities. Use the method of your...Ch. 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 - Assorted Probabilities. Use the method of your...Ch. 7.B - Prob. 52ECh. 7.B - Prob. 53ECh. 7.B - Assorted Probabilities. Use the method of your...Ch. 7.B - Assorted Probabilities. Use the method of your...Ch. 7.B - 56. How Many Rolls? At least how many times do you...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 - Lottery Odds. The probability of a 2 winner in a...Ch. 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 - You are betting on a game in which each bet has an...Ch. 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 - Cameron is betting on a game in which the...Ch. 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 - Expected Value in Games. Find the expected value...Ch. 7.C - Expected Value Games. Find the expected value (to...Ch. 7.C - Expected Value in Games. Find the expected value...Ch. 7.C - Expected Value Games. Find the expected value (to...Ch. 7.C - Insurance Claims. Find the expected value (to the...Ch. 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 - 26. Baseball Batting Averages. Based on his record...Ch. 7.C - Prob. 27ECh. 7.C - Prob. 28ECh. 7.C - Prob. 29ECh. 7.C - Prob. 30ECh. 7.C - equation reference goes here31. House Edge in...Ch. 7.C - Prob. 32ECh. 7.C - Powerball. The table below gives prizes and...Ch. 7.C - Prob. 34ECh. 7.C - Prob. 35ECh. 7.C - 35-36: Mega Millions. The table below gives prizes...Ch. 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. 40ECh. 7.C - Prob. 41ECh. 7.C - Prob. 42ECh. 7.C - The Morality of Gambling. Republican Senator...Ch. 7.C - Prob. 44ECh. 7.D - Prob. 1QQCh. 7.D - Prob. 2QQCh. 7.D - Prob. 3QQCh. 7.D - Prob. 4QQCh. 7.D - 5. Based on Table 7.5 and a U.S. population of 325...Ch. 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 - 9-12: Data in Perspective. Use these data to help...Ch. 7.D - 9-12: Data in Perspective. Use these data to help...Ch. 7.D - Prob. 11ECh. 7.D - Prob. 12ECh. 7.D - 13. Twenty-Year Trend in Automobile Safety. Use...Ch. 7.D - Prob. 14ECh. 7.D - Prob. 15ECh. 7.D - Prob. 16ECh. 7.D - Prob. 17ECh. 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 - 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 - U.S. Birth and Death Rates. In 2015, the U.S....Ch. 7.D - Prob. 30ECh. 7.D - 31. Aging population. The table shows the total...Ch. 7.D - 32. Psychology of Expected Values. The...Ch. 7.D - 33. Car Safety in Perspective. The opening...Ch. 7.D - Prob. 34ECh. 7.D - Prob. 35ECh. 7.D - Prob. 36ECh. 7.D - Prob. 37ECh. 7.D - Prob. 38ECh. 7.D - Risk in the News. Find a recent news report...Ch. 7.E - You are asked to create a 5-character password,...Ch. 7.E - A waitress has four different entrees for the four...Ch. 7.E - A teacher has 28 students. and 5 them will be...Ch. 7.E - The number of permutations of 12 objects grouped 5...Ch. 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 - Counting Methods. Answer the following questions...Ch. 7.E - Counting Methods. Answer the following questions...Ch. 7.E - Counting Methods. Answer the following questions...Ch. 7.E - Counting Methods. Answer the following questions...Ch. 7.E - Counting Methods. Answer the following questions...Ch. 7.E - Counting Methods. Answer the following questions...Ch. 7.E - Prob. 29ECh. 7.E - Counting Methods. Answer the following questions...Ch. 7.E - Counting Methods. Answer the following questions...Ch. 7.E - 23-40: Counting Methods. Answer the following...Ch. 7.E - Counting Methods. Answer the following questions...Ch. 7.E - Counting Methods. Answer the following questions...Ch. 7.E - Counting Methods. Answer the following questions...Ch. 7.E - Counting Methods. Answer the following questions...Ch. 7.E - Counting Methods. Answer the following questions...Ch. 7.E - Counting Methods. Answer the following questions...Ch. 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 - Counting and Probability. Find the probability of...Ch. 7.E - Counting and Probability. Find the probability of...Ch. 7.E - Counting and Probability. Find the probability of...Ch. 7.E - Counting and Probability. Find the probability of...Ch. 7.E - 47-54: Counting and Probability. Find the...Ch. 7.E - Counting and Probability. Find the probability of...Ch. 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. 64ECh. 7.E - Prob. 65E
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- 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
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