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4.6: Practice (Chapter 4)

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    4.1: What is Probability?

    1. Define probability in your own words.
    2. Why do we refer to probability as a “long-run” relative frequency?
    3. What does the Law of Large Numbers say about long-run probabilities as the number of trials increases?
    4. Why is randomness important in interpreting statistical results?
    5. Describe a real-life context where probability helps people make better decisions.
    6. For the following problems consider flipping a fair coin with heads on one side and tails on the other.
      1. What is the theoretical probability of the coin landing heads up?
      2. Flip a fair coin 20 times and record your results. What proportion of the flips were heads?
      3. Simulate 100 coin flips (you can use an applet or spreadsheet). How did the proportion of heads compare to the theoretical probability?
      4. 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
      5. How many times could we get the same number from rolling a die before we start to think it is unfair? 
    7. For the following problems consider rolling a fair six sided die numbered one through six.
      1. What is the probability of rolling a 5?
      2. What is the probability of rolling an even number?
      3. What is the probability of rolling a prime number?
      4. Roll a die 20 times and record your results. What proportion of the rolls were 5's?
      5. Simulate 100 die rolls (you can use an applet or spreadsheet). How did the proportion of 5's compare to the theoretical probability?
      6. If a fair die is rolled many times and the last ten rolls are all 5's, then is the chance that the next roll will be a 5 somewhat less than 16.6%? Explain
    8. For the following problems consider drawing a card from a standard deck of playing cards.
      1. What is the theoretical probability of drawing a red card?
      2. What is the theoretical probability of drawing a club?
      3. What is the theoretical probability of drawing a face card (jack, queen, or king)?
      4. How does shuffling the deck of cards affect the probability of drawing a specific card?
    9. Explain what is wrong with the following statements. Use complete sentences.
      1. If there is a 60% chance of rain on Saturday and a 70% chance of rain on Sunday, then there is a 130% chance of rain over the weekend.
      2. The probability that a baseball player hits a home run is greater than the probability that he gets a successful hit.

    4.2: Introduction to Counting

    1. How many different outcomes are there for flipping a coin 23 times?
    2. How many different outcomes are there for rolling a die 100 times?
    3. How many different outcomes are there for drawing five cards from a shuffled deck of playing cards?
    4. 5 different color dice are rolled, and the numbers showing are recorded. How many different outcomes are possible?
    5. You’re choosing an outfit from 3 shirts, 2 pants, and 2 pairs of shoes. How many different outfits are possible?
    6. 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 different flags can be designed?
    7. A restaurant offers 5 appetizers and 4 entrees. If you order one of each, how many meal combinations are available?
    8. 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?
    9. Compute 5!
    10. Write the following in factorial notation 1x2x3x...x22x23
    11. Simplify and then compute 5!/3!
    12. Simplify and then compute 11!/(9!2!)
    13. Write the following using factorial notation 7x8x9X10
    14. The Colorado Lottery Mega Millions has players choose five numbers from 1 to 70 and one number from 1 to 24. How many different lottery tickets are there?
    15. How many 3-letter codes can be made from A–Z with no repetition?
    16. How many different ways can the 7 main Harry Potter books be arranged on a shelf?
    17. The standard Colorado license plate includes seven characters. These can be: letters (A-Z), digits (1-9), blank space, dash, and period. How many different CO license plates are there?
    18. In how many ways can first, second, and third prizes be awarded in a contest with 590 contestants?
    19. A committee consists of 13 people. How many different subcommittees of four people can be formed?
    20. You are selecting 3 students from a class of 10 to represent the class. How many different combinations of students are possible?
    21. 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?
    22. A vending machine has 8 different soft drink options. How many pairs of drinks can you buy?
    23. Explain the difference between permutations and combinations. Give an example of when each would be appropriate.

    4.3: Foundational Probability Rules

    1. An experiment consists of first rolling a die and then tossing a coin.
      1. List the sample space.
      2. List the outcomes for the event described by rolling a number greater than 4
    2. An experiment consists of measuring the weight of a dog in a shelter. Describe the sample space.
    3. What is the event set for rolling a pair of dice and getting the same number for both?
    4. Two Queen cards are dealt from a standard deck of playing cards. Write out the event set.
    5. What is the relationship between experiments and events?
    6. What is the relationship between outcomes and sample space?
    7. When is the sample space the same as the event space?
    8. 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, and let S = the event that a country is in South America.
      1. Find P(A)
      2. Find P(E)
      3. Find P(F)
      4. Find P(N)
      5. Find P(O)
      6. Find P(S)
    9. A special deck of cards has ten cards. Four are green, three are blue, and three are red. When a card is picked, its color of it is recorded. An experiment consists of first picking a card and then tossing a coin.
      1. List the sample space.
      2. Let A be the event that a blue card is picked first, followed by landing a head on the coin toss. Find P(A).
      3. Let B be the event that a red or green is picked, followed by landing a head on the coin toss. Are the events A and B mutually exclusive? Explain your answer in one to three complete sentences, including numerical justification.
      4. Let C be the event that a red or blue is picked, followed by landing a head on the coin toss. Are the events A and C mutually exclusive? Explain your answer in one to three complete sentences, including numerical justification.
    10. When rolling a pair of dice, is getting a total of 7 mutually exclusive with rolling a pair of the same number? Explain
    11. Are rain and sunny weather mutually exclusive? Explain
    12. Are drawing a face card (jack, queen, or king) and drawing a red card from a full deck of playing cards mutually exclusive events? Explain
    13. Are drawing a face card and drawing an ace from a full deck of playing cards mutually exclusive events? Explain
    14. 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.
    15. Make a statement in ordinary English that describes the complement of each event (do not simply insert the word “not”).
      1. In the roll of a die: “five or more.”
      2. In a roll of a die: “an even number.”
      3. In two tosses of a coin: “at least one heads.”
      4. In the random selection of a college student: “Not a first-year student.”
    16. Use the complement rule to show that the probability of flipping a coin and not getting heads is 0.5.
    17. Draw a Venn diagram showing two overlapping events A and B. Shade in the area that corresponds to the BC.
    18. Write the Addition Rule for any two events A and B.
    19. If P(A) = 0.4, P(B) = 0.5, and A and B are mutually exclusive, what is P(A ∪ B)?
    20. Use the addition rule to show that the probability of flipping a coin and getting heads or tails is 1.
    21. Compute the following probabilities for rolling a fair die:
      1. P(5C)
      2. P(even or 2)
      3. P(prime or odd)
      4. P(<2 or >4)
    22. Compute the following probabilities for drawing a card from a standard deck of playing cards:
      1. P(heartC)
      2. P(red or king)
      3. P(ace or club)
      4. P(diamond or face card)
      5. P(black or spade)
      6. P(J or Q or K)

    23.  The casino game, roulette, allows the gambler to bet on the probability of a ball, which spins in the roulette wheel, landing on a particular color, number, or range of numbers. The table used to place bets contains of 38 numbers, and each number is assigned to a color and a range.

      This is an image of a roulette table.
      1. List the sample space of the 38 possible outcomes in roulette.
      2. Find P(red)
      3. Find P(1st Dozen)
      4. Find P(even or 1 to 18)
      5. Find P((0 or 00)C)
      6. Is getting an odd number the complement of getting an even number? Why?
      7. Find two mutually exclusive events.

    24. The graph shown is based on more than 170,000 interviews done by Gallup that took place from January through December 2012. The sample consists of employed Americans 18 years of age or older. The Emotional Health Index Scores are the sample space. We randomly sample one Emotional Health Index Score.

      emotional health index score
      Figure 3.3.1.
      1. Find the probability that an Emotional Health Index Score is 82.7.
      2. Find the probability that an Emotional Health Index Score is 81.0.
      3. Find the probability that an Emotional Health Index Score is more than 81?
      4. Find the probability that an Emotional Health Index Score is between 80.5 and 82?
      5. If we know an Emotional Health Index Score is 81.5 or more, what is the probability that it is 82.7?
      6. What is the probability that an Emotional Health Index Score is 80.7 or 82.7?
      7. What is the probability that an Emotional Health Index Score is less than 80.2 given that it is already less than 81.
      8. What occupation has the highest emotional index score?
      9. What occupation has the lowest emotional index score?
      10. What is the range of the data?
      11. Compute the average EHIS.
      12. If all occupations are equally likely for a certain individual, what is the probability that they will have an occupation with lower than average EHIS?

    4.4: Compound Events, Independence, and Conditional Probability

    1. What is a compound event? Provide one example from daily life.
    2. Define conditional probability and explain how it differs from basic probability.
    3. A die is rolled and then a coin is flipped. Are these events independent? Why?
    4. Differentiate between dependent and independent events. Provide one example of each.
    5. In your own words, explain the difference between mutually exclusive and independent events.
    6. 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.
    7. Repeat the previous question, but assume you replace the first marble before drawing the second. How does the probability change?
    8. Suppose you roll two standard 6-sided dice. Event A is the total of the two dice is 7. Event B is that the first die is even. Are events A and B independent?
    9. When is drawing a second card from a deck independent of the first?
    10. In a group of 100 people, 30 have the flu, and 24 of those tested positive. Write and calculate P(Test Positive | Has Flu).
    11. 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?
    12. Suppose 80% of people like peanut butter, 89% like jelly, and 78% like both. Given that a randomly sampled person likes peanut butter, what's the probability that they also like jelly?
    13. If P(A) = 0.6 and P(B) = 0.3, and A and B are independent, calculate P(A ∩ B).
    14. Let P(A ∩ B) = 0.1 and P(B) = 0.2. Find P(A | B).
    15. Using the formula for conditional probability, explain what each component of P(A | B) = P(A ∩ B)/P(B) represents in words.
    16. If two events A and B are independent, how is conditional probability simplified?
    17. 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).
    18. Suppose we roll two fair dice. Each die has six faces.
      1. List the sample space.
      2. Let A be the event that either a three or four is rolled first, followed by an even number. Find P(A).
      3. Let B be the event that the sum of the two rolls is at most seven. Find P(B).
      4. Find P(A ∩ B).
      5. In words, explain what P(A|B) represents. Find P(A|B).
      6. Are A and B mutually exclusive events? Explain your answer in one to three complete sentences, including numerical justification.
      7. Are A and B independent events? Explain your answer in one to three complete sentences, including numerical justification.
      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.
      1. Are being Independent and being a swing voter disjoint, i.e. mutually exclusive?
      2. Draw a Venn diagram summarizing the variables and their associated probabilities.
      3. What percent of voters are Independent but not swing voters?
      4. What percent of voters are Independent or swing voters?
      5. What percent of voters are neither Independent nor swing voters?
      6. What percent of voters are Independent and swing voters?
      7. Is the event that someone is a swing voter independent of the event that someone is a political Independent?
    20. 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.
      1. Are living below the poverty line and speaking a language other than English at home disjoint?
      2. Draw a Venn diagram summarizing the variables and their associated probabilities.
      3. What percent of Americans live below the poverty line and only speak English at home?
      4. What percent of Americans live below the poverty line or speak a language other than English at home?
      5. What percent of Americans live above the poverty line and only speak English at home?
      6. 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?
    21. 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?

    Practice problems include parts of:

    Introductory Statistics 2e by OpenStax is licensed under CC BY 4.0

    Access for free at https://openstax.org/books/introductory-statistics-2e/pages/1-introduction

    and

    OpenIntro Statistics by  David Diez, Christopher Barr, & Mine Çetinkaya-Rundel is licensed under CC BY 3.0

    4.6: Practice (Chapter 4) is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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