4.6: Practice (Chapter 4)
- Page ID
- 59135
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)4.1: What is Probability
- Define probability in your own words. Why do we call it a “long-run” relative frequency?
- Flip a fair coin 20 times. Record your results. What proportion of the flips were heads?
- Simulate 100 coin flips (you can use an applet or spreadsheet). How did the proportion of heads compare to the theoretical probability?
- What does the Law of Large Numbers say about long-run probabilities as the number of trials increases?
- If a fair coin is tossed many times and the last eight tosses are all heads, then is the chance that the next toss will be heads is somewhat less than 50%? Explain
- How many times could we get the same number from rolling a die before we start to think it is unfair?
- Describe a real-life context where probability helps people make better decisions.
- Why is randomness important in interpreting statistical results?
4.2: Introduction to Counting
- Compute the following
- 23!
- 5!/3!
- 11!/(9!2!)
- You’re choosing an outfit from 3 shirts, 2 pants, and 2 pairs of shoes. How many different outfits are possible?
- Taco del Tierra sells only tacos. They offer four types of tortillas, two types of beans, four types of vegetables, and four types of salsa. If you must choose exactly one of each for your taco, how many different kinds of tacos can you order at Taco del Tierra?
- Suppose a designer has a palette of 6 colors to work with, and wants to design a flag with 4 vertical stripes, all of different colors.
- How many 3-letter codes can be made from A–Z with no repetition?
- How many different ways can 5 books be arranged on a shelf?
- In how many ways can first, second, and third prizes be awarded in a contest with 590 contestants?
- 5 different color dice are rolled, and the numbers showing are recorded. How many different outcomes are possible?
- Explain the difference between permutations and combinations. Give an example of when each would be appropriate.
- You are selecting 3 students from a class of 10 to represent the class. How many different combinations of students are possible?
- A restaurant offers 5 appetizers and 4 entrees. If you order one of each, how many meal combinations are available?
- Joni is reorganizing their video game collection and is having trouble determining how to arrange them on the shelf. The collection consists of 39 games. Which method of counting should be used to determine the number of different ways to arrange the games?
4.3: Foundational Probability Rules
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There are 23 countries in North America, 12 countries in South America, 47 countries in Europe, 44 countries in Asia, 54 countries in Africa, and 14 in Oceania (Pacific Ocean region).
Let A = the event that a country is in Asia.
Let E = the event that a country is in Europe.
Let F = the event that a country is in Africa.
Let N = the event that a country is in North America.
Let O = the event that a country is in Oceania.
Let S = the event that a country is in South America.- Find P(A)
- Find P(E)
- Find P(F)
- Find P(N)
- Find P(O)
- Find P(S)
- Write the Addition Rule for any two events A and B.
- In your own words, explain the difference between mutually exclusive and independent events.
- Is drawing a face card (jack, queen, or king) and drawing a red card from a full deck of playing cards mutually exclusive events? Explain
- Is drawing a face card and drawing an ace from a full deck of playing cards mutually exclusive events? Explain
- If P(A) = 0.4, P(B) = 0.5, and A and B are mutually exclusive, what is P(A ∪ B)?
- If P(A) = 0.6 and P(B) = 0.3, and A and B are independent, calculate P(A ∩ B).
- Let P(A ∩ B) = 0.1 and P(B) = 0.2. Find P(A | B).
- What does the complement rule say about P(Aᶜ)? Use it to calculate the probability that a randomly selected person does not own a pet if 72% of people do.
- Make a statement in ordinary English that describes the complement of each event (do not simply insert the word “not”).
- In the roll of a die: “five or more.”
- In a roll of a die: “an even number.”
- In two tosses of a coin: “at least one heads.”
- In the random selection of a college student: “Not a first-year student.”
- Draw a Venn diagram showing two overlapping events A and B. Label the intersection and explain what it represents.
4.4: Compound Events, Independence, and Conditional Probability
- What is a compound event? Provide one example from daily life.
- Two coins are flipped. What is the sample space? What is the probability that both come up heads?
- If you roll a pair of fair dice, what is the probability of
- getting a sum of 1?
- getting a sum of 5?
- getting a sum of 12?
- Differentiate between dependent and independent events. Provide one example of each.
- You randomly draw two marbles from a jar with 3 red and 2 blue marbles. Compute the probability that both are red if you do not replace the first marble.
- Repeat the previous question, but assume you replace the first marble before drawing the second. How does the probability change?
- A die is rolled and then a coin is flipped. Are these events independent? Why?
- Define conditional probability and explain how it differs from basic probability.
- In a group of 100 people, 30 have the flu, and 24 of those tested positive. Write and calculate P(Test Positive | Has Flu).
- A survey shows 60% of students have laptops. Of those, 40% also have tablets. What is the probability that a randomly selected student has both?
- Using the formula for conditional probability, explain what each component of P(A | B) = P(A ∩ B)/P(B) represents in words.
- If two events A and B are independent, how is conditional probability simplified?
- Use a tree diagram or two-way table to organize outcomes for students who pass/fail a quiz based on whether they studied. Use hypothetical values and calculate P(Pass | Studied).
- A 2012 Pew Research survey asked 2,373 randomly sampled registered voters their political affiliation (Republican, Democrat, or Independent) and whether or not they identify as swing voters. 35% of respondents identified as Independent, 23% identified as swing voters, and 11% identified as both.
- Are being Independent and being a swing voter disjoint, i.e. mutually exclusive?
- Draw a Venn diagram summarizing the variables and their associated probabilities.
- What percent of voters are Independent but not swing voters?
- What percent of voters are Independent or swing voters?
- What percent of voters are neither Independent nor swing voters?
- Is the event that someone is a swing voter independent of the event that someone is a political Independent?
- The American Community Survey is an ongoing survey that provides data every year to give communities the current information they need to plan investments and services. The 2010 American Community Survey estimates that 14.6% of Americans live below the poverty line, 20.7% speak a language other that English at home, and 4.2% fall into both categories.
- Are living below the poverty line and speaking a language other than English at home disjoint?
- Draw a Venn diagram summarizing the variables and their associated probabilities.
- What percent of Americans live below the poverty line and only speak English at home?
- What percent of Americans live below the poverty line or speak a language other than English at home?
- What percent of Americans live above the poverty line and only speak English at home?
- Is the event that someone lives below the poverty line independent of the event that the person speaks a language other than English at home?
- A genetic test is used to determine if people have a predisposition for thrombosis, which is the formation of a blood clot inside a blood vessel that obstructs the flow of blood through the circulatory system. It is believed that 3% of people actually have this predisposition. The genetic test is 99% accurate if a person actually has the predisposition, meaning that the probability of a positive test result when a person actually has the predisposition is 0.99. The test is 98% accurate if a person does not have the predisposition. What is the probability that a randomly selected person who tests positive for the predisposition by the test actually has the predisposition?