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Section3.3Chapter 3 Exercises

  1. List all possible outcomes of each of the following random experiments:

    1. A coin is tossed four times in a row. The observation is how the coin lands (H or T) on each toss.
    2. A student randomly guesses the answers to a four-question true-or-false quiz. The observation is the student's answer (T or F) for each question.
    3. A coin is tossed four times in a row. The observation is the percentage of tosses that are heads
    4. A student randomly guesses the answers to a four-question true-or-false quiz. The observation is the percentage of correct answers in the test.
  2. List all possible outcomes of each of the following random experiments:

    1. Roll three dice. The observation is the total of the three numbers rolled.
    2. Toss a coin five times. The observation is the difference (# of heads - # of tails) in the five tosses.
  3. Consider an experiment in which a fair coin is tossed once and a balanced die is rolled once. Describe the set of outcomes for this experiment. What is the probability that a head will be obtained on the coin and an odd number will be obtained on the die?
  4. A card is drawn at random out of a well-shuffled deck of 52 cards. Find the probability of each of the following events.

    1. Draw a queen.
    2. Draw a heart.
    3. Draw a face card. (A ''face'' card is a jack, queen, or king.)
  5. Robin bought a snack variety box containing 6 bags of chips, 8 bags of pretzels, and 4 bags of cookies. Each day, Robin reaches into the box and picks a snack at random. On the first day, Robin had chips. On the second and third days, Robin had pretzels. What is the probability that Robin will have cookies on the fourth day?
  6. If two balanced dice are rolled, what is the probability that the difference between the two numbers that appear will be less than 3?
  7. Jamie and Kyle ask a Magic 8-ball three questions. A Magic 8-ball has 20 possible responses for a question.

    1. What is the probability that the responses are ''Ask again later,'' ''It is decidedly so,'' and ''Very doubtful'' in that order?
    2. What is the probability that all three responses are ''Ask again later''?
    3. What is the probability that at least one response is ''Ask again later''?
  8. One ball is to be selected from a box containing red, white, blue, yellow, and green balls. If the probability that the selected ball will be red is 1/5 and the probability that it will be white is 2/5, what is the probability that it will be blue, yellow, or green?
  9. The University is trying a new policy in the Fall of 2019 to make moving into the dorms easier. Every student will be assigned a random day between Sunday and Thursday during the week before classes start. Students must move in on their assigned day to reduce congestion. For all students, the probability of being assigned each day is the same.

    1. What is the probability that you will be assigned to move in on Tuesday?
    2. Your friend Alice wants to move in on either Sunday, Monday, or Tuesday. What is the probability that she will be assigned one of those days?
    3. Bob and Charlie are brothers. They don't care what days they are assigned, as long as they are assigned the same day as each other. What is the probability they will get their wish?
  10. There are four teams playing in the September Silliness college basketball tournament: Nebraska (NE), Iowa (IA), Minnesota (MN), and Wisconsin (WI). In the first round, Nebraska plays Iowa and Minnesota plays Wisconsin. In the second round, the two teams who won the first round play each other, and the two teams who lost the first round play each other. Sportscasters predict that in the first round, NE has a 2/3 probability of beating IA and MN has a 3/5 probability of beating WI. In the second round, they predict that NE would have a 1/2 probability of winning against MN, and a 7/10 probability of winning against WI. Suppose that no game can end in a tie.

    1. Are the results of the two first round games dependent or independent events?
    2. You trust the sportscasters and fill out a bracket predicting that NE and MN will win the first round. What is the probability that at least part of your prediction turns out to be wrong?
    3. Are Nebraska's results in the first and second games dependent or independent events?
    4. Given that NE wins the first game, what is the probability that NE wins the second game? Are Nebraska's results in the first and second games dependent or independent events?
    5. What is the probability that NE wins the tournament? (That is, what is the probability that NE wins the first game and then wins the second game?)
  11. A group of people is introducing themselves to each other at the beginning of a conference. Assume throughout the problem that everyone has a name spelled using the 26 letters of the English alphabet, and that each letter is equally likely to appear.

    1. Two people are the first to sit down at their table. What is the probability that their first names do not start with the same letter?
    2. They are joined by a third person. What is the probability that the three persons' first names all start with different letters?
    3. What is the probability that there is at least one pair of the three people who have first names starting with the same letter?
  12. Crown and Anchor is a simple dice game that was once popular among sailors in the British navy and is still played on occasion, being one of the events at the Battle of Flowers Funfair, an annual carnival held on the island of Jersey in the North Sea. The game is played using three six-sided dice marked with distinctive symbols, including a red crown, a black anchor, and the four symbols used to mark suits in a standard deck of playing cards. Players bet $1 on one of the symbols, roll the three dice, and receive a payoff based on the number of dice that match their chosen symbol. Results are summarized in the table below.

    1. Find the expected value of a play of the Crown and Anchor game. You can use either the actual probabilities or a set of counts chosen to match the actual probabilities.
    2. On the average, how much should you expect to lose at Crown and Anchor for every dollar that you bet?
Number of dice that match the bet 0 1 2 3
Payout per $1 bet $0 $2 $3 $4
Probability 125/216=0.5787 75/216=.3472 15/216=.0694 1/216=.0046
Table3.12Crown and Anchor Results