MATH 384. Guía 5. 1. An experiment involves tossing a single die. These are some events: A: Observe a 2 B: Observe an even number C: Observe a number greater than 2 D: Observe both A and B E: Observe A or B or both F: Observe both A and C a. List the simple events in the sample space. b. List the simple events in each of the events A through F. c. What probabilities should you assign to the simple events? d. Calculate the probabilities of the six events A through F by adding the appropriate simple-event probabilities. 2. A sample space S consists of five simple events with these probabilities: P(E1)=P(E2)=0.15; P(E3)=0.4; P(E4)=2P(E5) a. Find the probabilities for simple events E4 and E5 b. Find the probabilities for these two events: A: EI, E3, E4 B: E2, E3 c. List the simple events that are either in event A or event B or both. d. List the simple events that are in both event A and event B. 3. A sample space contains 10 simple events: EI,E2. . . . , E10. If P(E1) = 3P(E2) = 0.45 and the remaining simple events are equiprobable, find the probabilities of these remaining simple events. 4. A particular basketball player hits 70% of her free throws. When she tosses a pair of free throws, the four possible simple events and three of their associated probabilities are as given in the table: Simple Outcome of Outcome of Event First Free Throw Second Free Throw Probability _______________________________________________________ 1 Hit Hit .49 2 Hit Miss ? 3 Miss Hit .21 4 Miss Miss .09 a) Find the probability that the player will hit on the forst trowand miss in the second. b) Find the probability that the player will hit on at least one of the two free trows. 5. A survey classified a large number of adults according to whether they were judged to need eyeglasses to correct their reading vision and whether they used eyeglasses when reading. The proportions falling into the four categories are shown in the table. (Note that a small proportion, .02, of adults used eyeglasses when in fact they were judged not to need them.) Used Eyeglasses for Reading Judged to Need Eyeglasses Yes No Yes 0.44 0.14 No 0.02 0.40 If a single adult is selected from this large group, find the probability of each event: a. The adult is judged to need eyeglasses. b. The adult needs eyeglasses for reading but does not use them. c. The adult uses eyeglasses for reading whether he or she needs them or not. 6. The game of roulette uses a wheel containing 38 pockets. Thirty-six pockets are numbered 1,2, . . .,36, and the remaining two are marked 0 and 00. The wheel is spun, and a pocket is identified as the "winner." Assume that the observance of anyone pocket is just as likely as any other. a. Identify the simple events in a single spin of the roulette wheel. b. Assign probabilities to the simple events. c. Let A be the event that you observe either a 0 or a 00. List the simple events in the event A and find P(A). d. Suppose you placed bets on the numbers 1 through 18. What is the probability that one of your numbers is the winner? 7. Three people are randomly selected from voter registration and driving records to report for jury duty. The gender of each person is noted by the county clerk. a. Define the experiment. b. List the simple events in S. c. If each person is just as likely to be a man as a woman, what probability do you assign to each simple event? d. What is the probability that only one of the three is a man? e. What is the probability that all three are women? Refer to Exercise 7. Suppose that there are six prospective jurors, four men and two women, who might be impaneled to sit on the jury in a criminal case. Two jurors are randomly selected from these six to fill the two remaining jury seats. a. List the simple events in the experiment (HINT: There are 15 simple events if you ignore the order of selection of the two jurors.) b. What is the probability that both impaneled jurors are women? 8. A food company plans to conduct an experiment to compare its brand of tea with that of two competitors. A single person is hired to taste and rank each of three brands of tea, which are unmarked except for identifying symbols A, B, and C. a. Define the experiment. b. List the simple events in S. c. If the taster has no ability to distinguish difference in taste among teas, what is the probability that the taster will rank tea type A as the most desirable? As the least desirable? 9. Four equally qualified runners, John, Bill, Ed, and Dave, run a l00-meter sprint, and the order of finish is recorded. a. How many simple events are in the sample space? b. If the runners are equally qualified, what probability should you assign to each simple event? c. What is the probability that Dave wins the race? d. What is the probability that Dave wins and John places second? e. What is the probability that Ed finishes last? 10. In a genetics experiment, the researcher mated two Drosophila fruit flies and observed the traits of 300 offspring. The results are shown in the table. Eye Color Normal Vermillion | | | | Wing Size Normal Miniature 140 3 6 151 One of these offspring is randomly selected and observed for the two genetic traits. a. What is the probability that the fly has normal eye color and normal wing size? a. What is the probability that the fly has vermillion eyes? a. What is the probability that the fly has either vermillion eyes or miniature wings, or both? MATH 384. Guía 6. 11. You have two groups of distinctly different items, 10 in the first group and 8 in the second. If you select one item from each group, how many different pairs can you form? 12. You have three groups of distinctly different items, four in the fIrst group, seven in the second, and three in the third. If you select one item from each group, how many different triplets can you form? 13. Evaluate the following permutations. (HINT: Your scientific calculator may have a function that allows you to calculate permutations and combinations quite easily.) a. P35 b. P910 c. P66 d. P120 1. Evaluate these combinations: a. C35 b. C910 c. C66 d. C120 In how many ways can you select fIve people from a group of eight if the order of selection is important? In how many ways can you select two people from a group of 20 if the order of selection is not important? Three dice are tossed. How many simple events are in the sample space? Four coins are tossed. How many simple events are in the sample space? Three balls are selected from a box containing 10 balls. The order of selection is not important. How many simple events are in the sample space? You own 4 pairs of jeans, 12 clean T-shirts, and 4 wearable pairs of sneakers. How many outfits (jeans, T-shirt, and sneakers) can you create? A businessman in New York is preparing an itinerary for a visit to six major cities. The distance traveled, and hence the cost of the trip, will depend on the order in which he plans his route. How many different itineraries (and trip costs) are possible? Your family vacation involves a cross-country air flight, a rental car, and a hotel stay in Boston. If you can choose from four major air carriers, five car rental agencies, and three major hotel chains, how many options are available for your vacation accommodations? Three students are playing a card game. They decide to choose the fmt person to play by each selecting a card from the 52-card deck and looking for the highest card in value and suit. They rank the suits from lowest to highest: clubs, diamonds, hearts, and spades. a. If the card is replaced in the deck after each student chooses, how many possible configurations of the three choices are possible? b. How many configurations are there in which each student picks a different card? c. What is the probability that all three students pick exactly the same card? d. What is the probability that all three students pick different cards? A French restaurant in Riverside, California, offers a special summer menu in which, for a fixed dinner cost, you can choose from one of two salads, one of two entrees, and one of two desserts. How many different dinners are available? Five cards are selected from a 52-card deck for a poker hand. a. How many simple events are in the sample space? b. A royal flush is a hand that contains the A, K, Q, J, and 10, all in the same suit. How many ways are there to get a royal flush? c. What is the probability of being dealt a royal flush? Refer to Exercise (five cards). You have a poker hand containing four of a kind. a. How many possible poker hands can be dealt? b. In how many ways can you receive four cards of the same face value and one card from the other 48 available cards? c. What is the probability of being dealt four of a kind? A study is to be conducted in a hospital to determine the attitudes of nurses toward various administrative procedures. If a sample of 10 nurses is to be selected from a total of 90, how many different samples can be selected? (HINT: Is order important in determining the makeup of the sample to be selected for the survey?) Two city council members are to be selected from a total of five to form a subcommittee to study the city's traffic problems. a. How many different subcommittees are possible? b. If all possible council members have an equal chance of being selected, what is the probability that members Smith and Jones are both selected? MATH 384. Guía 7. Identify the following as discrete or continuous random variables: a. Increase in length of life attained by a cancer patient as a result of surgery b. Tensile breaking strength (in pounds per square inch) of 1-inch-diameter steel cable c. Number of deer killed per year in a state wildlife preserve d. Number of overdue accounts in a department store at a particular time e. Your blood pressure A random variable x has this probability distribution: x P(x) 0 .1 1 .3 2 .4 3 .1 4 ? 5 .05 a. Find p(4). b. Construct a probability histogram to describe p(x). c. Find , 2 and . d. Locate the interval 2 on the x-axis of the histogram. What is the probability that x will fall into this interval? e. If you were to select a very large number of values of x from the population, would most fall into the interval 2 . Explain. A random variable x can assume five values: 0, I, 2, 3,4. A portion of the probability distribution is shown here: x 0 1 2 3 4 P(x) .1 .3 .3 ? .1 a. Find p(3). b. Construct a probability histogram for p(x). c. Calculate the population mean, variance, and standard deviation. d. What is the probability that x is greater than 2? e. What is the probability that x is 3 or less? Let x equal the number observed on the throw of a single balanced die. a. Find and graph the probability distribution for x. b. What is the average or expected value of x? c. What is the standard deviation of x? d. Locate the interval 2 on the x-axis of the graph in part a. What proportion of all the measurements would fall into this range? Let x represent the number of times a customer visits a grocery store in a 1-week period. Assume this is the probability distribution of x: x 0 1 2 3 P(x) .1 .4 .4 .1 Find the expected value of x, the average number of times a customer visits the store. Who is the king of late night TV? An Internet survey estimates that, when given a choice between David Letterman and Jay Leno, 52% of the population prefers to watch Jay Leno. Suppose that