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Finite Mathematics and Its Applications 12th edition Larry J. Goldstein, David I. Schneider, Martha J. Siegel, Steven Hair - Solutions

The next table shows the results from The American Freshman: National Norms Fall 2015 of a question posed to 141,000 college freshmen. Consider the experiment of selecting a student at random from the group surveyed and observing his or her answer. Determine the probability distribution for this
1. The following table shows the probability distributions of letter grades from a mathematics class. What is the probability that a randomly chosen student received a letter grade higher than F but lower than A? Letter Grade _____________ Probability A .................................... .29 B
An experiment with outcomes s1, s2, s3, s4, s5, s6 has the following probability distribution: Outcome ______________ Probability s1 .................................. .05 s2 .................................. .25 s3 .................................. .05 s4 ..................................
The table that follows was derived from a survey of college freshmen in 2015. Each probability is the likelihood that a randomly selected freshman applied to the specified number of colleges. For instance, 10% of the freshmen applied to just one college, and therefore, the probability that a
The next table summarizes the age distribution for a company's employees. Each probability is the likelihood that a randomly selected employee is in the specified age group. (a) Convert this data into a probability distribution with outcomes 20-34 years, 35-49 years, 50-64 years, and 65-79
1. Which of the following probabilities are feasible for an experiment having sample space {s1, s2, s3}? Explain your answer. (a) Pr (s1) = .4, Pr (s2) = .4, Pr (s3) = .4 (b) Pr (s1) = .5, Pr (s2) = .7, Pr (s3) = -.2 (c) Pr (s1) = 1/5, Pr (s2) = 2/5, Pr (s3) = 1/5 2. Which of the following
1. Three cars, a Mazda, a Honda, and a Ford, are in a quarter-mile race. The probability that the Mazda will win the race is 2/3, and the probability that the Honda will win is 1/4. Assuming no ties are possible, what is the probability that the Ford will win the race? 2. In a study, the residents
1. The probability that Alice beats Ben in a game of tennis is twice the probability that Ben beats Alice. Determine the two probabilities? 2. Suppose that a pair of dice is rolled. Find Pr (sum of the two numbers is odd) + Pr (sum of the two numbers is even).
1. An experiment consists of tossing a coin five times and observing the sequence of heads and tails. Find Pr (an even number of heads occurs) + Pr (an odd number of heads occurs)? 2. Suppose that Pr (E) = .4 and Pr (F) = .5, where E and F are mutually exclusive. Find Pr (E ( F)? 3. Suppose that Pr
Consider the probabilities shown in the Venn diagram in Figure 3.Figure 3:1. Determine the probability that the event E occurs. 2. Determine the probability that exactly one of the events E or F occurs. 3. Determine the probability that F occurs, but E does not occur. 4. Determine the probability
Use a Venn diagram similar to the one in Fig. 1 to solve the problem. 1. Suppose that Pr (E) = .6, Pr (F) = .5, and Pr (E ( F) = .4. Find (a) Pr (E ( F) (b) Pr (E ( F'). 2. Suppose that Pr (E) = .6, Pr (F) = .5, and Pr (E' ( F') = .6. Find (a) Pr (E ( F') (b) Pr (E' (F')?
1. Convert the odds of "10 to 1" to a probability. 2. Convert the odds of "4 to 5" to a probability. 3. Convert the probability .2 to odds. 4. Convert the probability 3/7 to odds?
1. The probability of getting three heads in five tosses of a coin is .3125. What are the odds of getting three heads? 2. The probability that a graduate of a Big Ten school will eventually earn a Ph.D. degree is .05. What are the odds of a Big Ten graduate eventually earning a Ph.D. degree? 3. The
1. Gamblers usually give odds against an event happening. For instance, if a bookie gives the odds 4 to 1 that the Yankees will win the next World Series, he is stating that the probability that the Yankees will win is 1/5, or .2. Also, if a bettor bets $1 that the Yankees will win and the Yankees
Determine the probability distribution for the given experiment. 1. Toss an unbiased coin twice, and count the number of heads. 2. A box contains seven slips of paper: one with a letter A printed on it, one with a B, three with a C, and two with a D. Draw a slip of paper and observe which letter is
Can be answered without any computations. 1. A high school astrology club has 13 members. What is the probability that two or more members have the same zodiac sign? Note: There are 12 zodiac signs. 2. Six people check their coats at a restaurant's coat-check counter. If the attendant returns the
1. The modern American roulette wheel has 38 slots, which are labeled with 36 numbers evenly divided between red and black, plus two green numbers 0 and 00. What is the probability that the ball will land on a green number? 2. A state is selected at random from the 50 states of the United States.
A number is chosen at random from the whole numbers between 1 and 17, inclusive. (a) What is the probability that the number is odd? (b) What is the probability that the number is even? (c) What is the probability that the number is a multiple of 3? (d) What is the probability that the number is
1. The U.S. Senate consists of two senators from each of the 50 states. Five senators are to be selected at random to form a committee. What is the probability that no two members of the committee are from the same state? 2. A factory produces LCD panels, which are packaged in boxes of 10. Three
Refer to a classroom of children (12 boys and 10 girls) in which seven students are chosen to go to the blackboard. 1. What is the probability that no boys are chosen? 2. What is the probability that the first three children chosen are boys? 3. What is the probability that at least two girls are
1. Three people are chosen at random. What is the probability that at least two of them were born on the same day of the week? 2. Four people are chosen at random. What is the probability that at least two of them were born in the same month? Assume that each month is as likely as any other.
1. Without consultation with each other, each of four organizations announces a one-day convention to be held during June. Find the probability that at least two organizations specify the same day for their convention. 2. There were 16 presidents of the Continental Congress from 1774 to 1788. Each
1. A number is chosen at random from the whole numbers between 1 and 100, inclusive. (a) What is the probability that the number ends in a zero? (b) What is the probability that the number is odd? (c) What is the probability that the number is odd or ends in a zero? 2. An urn contains five red
1. What is the probability that, in a group of 25 people, at least one person has a birthday on June 13? Why is your answer different from the probability displayed in Table 1 for r = 25? 2. Johnny Carson, host of The Tonight Show from 1962-1992, discussed the birthday problem during one of his
1. A die is rolled twice. What is the probability that the two numbers are different? 2. A die is rolled three times. What is the probability of obtaining three different numbers? 3. A die is rolled four times. What is the probability of obtaining only even numbers? 4. A die is rolled three times.
1. A coin is tossed 10 times. What is the probability of obtaining four heads and six tails? 2. A coin is tossed seven times. What is the probability of obtaining five heads and two tails? 3. A university admissions office randomly assigns each student from a group of four incoming freshmen to an
Figure 1 shows a partial map of the streets in New York City. (Such maps are discussed in Chapter 5.) A tourist starts at point A and selects at random a shortest path to point B. That is, they walk only south and east. Find the probability that(a) They pass through point C.(b) They pass through
Repeat Exercise 33 for Fig. 2.Figure 1 shows a partial map of the streets in New York City. (Such maps are discussed in Chapter 5.) A tourist starts at point A and selects at random a shortest path to point B. That is, they walk only south and east. Find the probability that(a) They pass through
1. In the American League, the East, Central, and West divisions each consists of five teams. A sportswriter predicts the winner of each of the three divisions by choosing a team completely at random in each division. What is the probability that the sportswriter will predict at least one winner
1. Suppose that the sportswriter in Exercise 36 simply puts the 12 team names in a hat and draws 3 completely at random. Does this increase or decrease the writer's chance of picking at least one winner? 2. Fred has five place settings consisting of a dinner plate, a salad plate, and a bowl. Each
1. An urn contains seven green balls and five white balls. A sample of three balls is selected at random from the urn. Find the probability that (a) Only green balls are selected. (b) At least one white ball is selected. 2. An urn contains six green balls and seven white balls. A sample of four
A man, a woman, and their three children randomly stand in a row for a family picture. What is the probability that the parents will be standing next to each other?
What is the probability that a random arrangement of the letters in the word GEESE has all the E's adjacent to one another?
Full house (three cards of one rank and two cards of another rank) A poker hand consists of five cards drawn from a deck of 52 cards. Each card has one of 13 ranks (2, 3, 4, . . . , 10, jack, queen, king, ace) and one of four suits (spades, hearts, diamonds, clubs). In Exercises 43-46, determine
Three of a kind (three cards of one rank and two cards of distinct rank, both different from the rank of the triple) A poker hand consists of five cards drawn from a deck of 52 cards. Each card has one of 13 ranks (2, 3, 4, . . . , 10, jack, queen, king, ace) and one of four suits (spades, hearts,
Two pairs (two cards of one rank, two cards of a different rank, and one card of a rank other than those two ranks) A poker hand consists of five cards drawn from a deck of 52 cards. Each card has one of 13 ranks (2, 3, 4, . . . , 10, jack, queen, king, ace) and one of four suits (spades, hearts,
One pair (two cards of one rank and three cards of distinct ranks, where each of the three cards has a different rank from the rank of the pair) A poker hand consists of five cards drawn from a deck of 52 cards. Each card has one of 13 ranks (2, 3, 4, . . . , 10, jack, queen, king, ace) and one of
1. A bridge hand consists of thirteen cards drawn from a deck of 52 cards. Each card has one of 13 ranks (2, 3, 4, . . . , 10, jack, queen, king, ace) and one of four suits (spades, hearts, diamonds, clubs). What is the probability of each of the following suit distributions in a bridge hand? (a)
1. What is the probability of winning the Illinois Lottery Lotto with a $1 bet? 2. In the game week ending June 18, 1983, a total of 2 million people bought $1 tickets, and 78 people matched all six winning integers and split the jackpot. If all numbers were selected randomly, the likelihood of
1. In the California Fantasy 5 lottery, a player pays $1 for a ticket and selects 5 numbers from the numbers 1 through 39. If they match exactly three of the five numbers drawn, they receive $15. What is the probability of selecting exactly three of the five numbers drawn? 2. The winning
1. Suppose that a study produced the following results: Out of a group of 60 people who took Math Helper before their math exam, only 8 of them failed the exam. Out of a group of 60 people who took a placebo before the exam, 14 of them failed the exam. Calculate the absolute and relative risk
1. Table 3 shows the experiences of 200 people who took a medication designed to prevent a certain condition. Calculate the absolute and the relative risk reductionTabledue to taking the medication. Also, calculate the number of people who must take the medication in order for one person to be
1. Suppose that 80 people must take a certain medication in order for one person to be helped. What is the absolute risk reduction of the intervention? 2. License Plate Game Johnny and Doyle are driving on a lightly traveled road. Johnny proposes the following game: They will look at the license
An urn contains eight red balls and six white balls. A sample of three balls is selected at random from the urn. Find the probability that (a) The three balls have the same color. (b) The sample contains more white balls than red balls?
Find the probability that at least two people in a group of size n = 5 select the same card when drawing from a 52-card deck with replacement. Determine the group size n for which the probability of such a match first exceeds 50%?
1. A political science class has 20 students, each of whom chooses a topic from a list for a term paper. How big a pool of topics is necessary for the probability of at least one duplicate to drop below 50%? 2. Refer to Exercise 24. How large would the audience have to have been so that the
1. A year on planet Ork has 100 days. Find the smallest number of Orkians for which the probability that at least two of them have the same birthday is 50% or more. 2. In many state lotteries, six numbers are selected from a set of numbers. Quite often, the winning selection contains two
1. Two out of the seven members of a school board feel that all high school students should be required to take a course in coding. A pollster selects three members of the board at random and asks them for their opinion on requiring a coding course. What is the probability that at least one of the
The Venn diagram in Fig. 3 shows the probabilities for its four basic regions. Find(a) Pr (E)(b) Pr (F)(c) Pr (E | F)(d) Pr (F | E).Figure 3:
1. A coin is tossed three times. What is the probability that the outcome contains no heads, given that exactly one of the coins shows a tail? 2. Bag of Marbles A bag contains five red marbles and seven white marbles. If a sample of four marbles contains at least one white marble, what is the
1. Suppose a family has two children and the youngest is a girl. What is the probability that both children are girls? 2. Suppose a family has two children and at least one is a girl. What is the probability that both children are girls? 3. Value of College Twenty-five percent of individuals in a
1. Sixty percent of the teachers at a certain high school are female. Forty percent of the teachers are females with a master's degree. What is the probability that a randomly selected teacher has a master's degree, given that the teacher is female?2. Table 1 shows the projected number of advanced
The Venn diagram in Fig. 4 shows the probabilities for its four basic regions. Find(a) Pr (E)(b) Pr (F)(c) Pr (E | F)(d) Pr (F | E).Figure 4
Table 2 shows the number of registered voting-age U.S. citizens (in millions) by gender and their reported participation in the 2014 congressional election. Find the probability that a voting-age citizen selected at random(a) Voted.(b) Is male.(c) Is female, given that the citizen voted.(d) Voted,
Table 3 shows the numbers (in thousands) of officers and enlisted persons on active military duty on December 31, 2015. Find the probability that a person in the military selected at random is(a) An officer.(b) A Marine.(c) An officer in the Marines.(d) An officer, given that they are a Marine.(e)
Table 4 shows the probable field of study for 1500 freshman males and 1000 freshman females. Find the probability that a freshman selected at random(a) Intends to major in business.(b) Is female.(c) Is a female intending to major in business.(d) Is male, given that the freshman intends to major in
1. Each of three sealed opaque envelopes contains two bills. One envelope contains two $1 bills, another contains two $5 bills, and the third contains a $1 bill and a $5 bill. An envelope is selected at random, and a bill is taken from the envelope at random. If it is a $5 bill, what is the
1. A coin is tossed five times. What is the probability that heads appears on every toss, given that heads appears on the first four tosses? 2. A coin is tossed twice. What is the probability that heads appeared on the first toss, given that tails appeared on the second toss?
1. According to exit polling for the 2016 Missouri Republican primary election, 48% of the primary voters were women. Nine percent of the women polled voted for John Kasich. What is the probability that a randomly selected voter from the poll is a woman who voted for John Kasich? 2. Twenty percent
Let S be a sample space and E and F be events associated with S. Suppose that Pr (E) = .5, Pr (F) = .4, and Pr (E ( F) = .1. Calculate (a) Pr (E | F) (b) Pr (F | E) (c) Pr (E | F') (d) Pr (E' | F')
1. Suppose that your team is behind by two points and you have the ball on your opponent's court with a few seconds left in the game. You can try a two-point shot (probability of success is .48) or a three-point shot (probability of success is .29). Which choice gives your team the greater
Let E and F be events with P(E) = .4, Pr (F) = .5, and Pr (E ( F) = .7. Are E and F independent events?
Let E and F be events with P(E) = .2, Pr (F) = .5, and Pr (E ( F) = .6. Are E and F independent events?
1. Let E and F be independent events with P(E) = .5 and Pr (F) = .6. Find Pr (E ( F)? 2. Let E and F be independent events with P(E) = .25 and Pr (F) = .4. Find Pr (E ( F).
Assume that E and F are independent events. Use the given information to find Pr (F). 1. Pr (E) = .7 and Pr (F E) = .6. 2. Pr (E) = .4 and Pr (F' | E') = .3. 3. Pr (E') = .6. and Pr (E ( F) = .1. 4. Pr (E) = .8 and Pr (E ( F ) = .4.
Let S be a sample space and E and F be events associated with S. Suppose that Pr (E) = .6, Pr (F) = .3, and Pr (E ( F) = .2. Calculate (a) Pr (E | F) (b) Pr (F | E') (c) Pr (E | F') (d) Pr (E' | F')
1. Let A, B, and C be independent events with Pr (A) = .4, Pr (B) = .1, and Pr (C) = .2. Calculate Pr [(A ( B ( C)']. 2. Let A, B, and C be independent events with Pr (A) = .2, Pr (A ( B) = .12, and Pr (A ( C) = .06. Calculate Pr (B ( C)?
1. A sample of two balls is drawn from an urn containing two white balls and three red balls. Are the events "the sample contains at least one white ball" and "the sample contains balls of both colors" independent? 2. An urn contains two white balls and three red balls. A ball is withdrawn at
1. Roll a die, and consider the following two events: E = {2, 4, 6}, F = {3, 6}. Are the events E and F independent? 2. Roll a die, and consider the following two events: E = {2, 4, 6}, F = {3, 4, 6}. Are the events E and F independent? 3. Rolling Dice Roll a pair of dice, and consider the sum of
1. A doctor studies the known cancer patients in a certain town. The probability that a randomly chosen resident has cancer is found to be .001. It is found that 30% of the town works for Ajax Chemical Company. The probability that an employee of Ajax has cancer is equal to .003. Are the events
Let S be a sample space and E and F be events associated with S. Suppose that Pr (E) = 1/3, Pr (F) = 5/12, and Pr (E ( F) = 2/3. Calculate (a) Pr (E ( F) (b) Pr (E | F) (c) Pr (F | E)
A medical screening program administers three independent tests. Of the persons taking the tests, 80% pass test I, 75% pass test II, and 60% pass test III. A participant is chosen at random. (a) What is the probability that they will pass all three tests? (b) What is the probability that they will
1. A "true-false" exam has 10 questions. Assuming that the questions are independent and that a student is guessing, find the probability that they get 100%? 2. A TV set contains five circuit boards of type A, five of type B, and three of type C. The probability of failing in its first 5000 hours
1. The probability that a fisherman catches a tuna in any one excursion is .15. What is the probability that he catches a tuna on each of three excursions? On at least one of three excursions? 2. A baseball player's batting average changes every time he goes to bat and therefore should not be used
1. A basketball player makes each free-throw with a probability of .6 and is on the line for a one-and-one free throw. (That is, a second throw is allowed only if the first is successful.) Assume that the two throws are independent. What do you think is the most likely result: scoring 0 points, 1
Let S be a sample space and E and F be events associated with S. Suppose that Pr (E) = 1/2, Pr (F) = 1/3, and Pr (E ( F) = 7/12. Calculate (a) Pr (E ( F) (b) Pr (E | F) (c) Pr (F | E)
1. Free-Throws Consider Exercise 59, but let the probability of success be p, where 0 < p < 1. For what value of p will the probability of scoring 1 point be the same as the probability of scoring 2 points? 2. A biased coin shows heads with probability .6. What is the probability of obtaining the
1. Find that value of N for which the probability of winning in Exercise 58 at least once in N successive plays is about .5. 2. If you bet "even" in roulette, the probability of winning is 9/19. Find the smallest value of N for which the probability of winning at least once on the "even" bet in N
Let S be a sample space and E and F be events associated with S. Suppose that Pr (E) = .4, Pr (F | E) = .25, and Pr (F) = .3. Calculate (a) Pr (E ( F) (b) Pr (E ( F) (c) Pr (E | F) (d) Pr (E' ( F)
Let S be a sample space and E and F be events associated with S. Suppose that Pr (E) = .5, Pr (F | E) = .4, and Pr (F) = .3. Calculate (a) Pr (E ( F) (b) Pr (E ( F) (c) Pr (E | F) (d) Pr (E ( F')
1. When a pair of dice is rolled, what is the probability that the sum of the dice is 8, given that the outcome is not 7? 2. When a pair of dice is rolled, what is the probability that the sum of the dice is 5, given that exactly one of the dice shows a 3? 3. A coin is tossed three times. What is
Draw trees representing the sequence of experiments. 1. Experiment I is performed. Outcome a occurs with probability .4, and outcome b occurs with probability .6. Then experiment II is performed. Its outcome c occurs with probability .8, and its outcome d occurs with probability .2. 2. Experiment I
A card is drawn from a 52-card deck. We continue to draw until we have drawn a king or until we have drawn five cards, whichever comes first. Draw a tree diagram that illustrates the experiment. Put the appropriate probabilities on the tree. Find the probability that the drawing ends before the
An urn contains six white balls and two red balls. Balls are selected one at a time (without replacement) until a white ball is selected. Find the probability that the number of balls selected is (a) one, (b) two, (c) three?
1. Twenty percent of the library books in the fiction section are worn and need replacement. Ten percent of the nonfiction holdings are worn and need replacement. The library's holdings are 40% fiction and 60% nonfiction. Use a tree diagram to find the probability that a book chosen at random from
1. Color blindness is a gender-linked inherited condition that is much more common among men than women. Suppose that 8% of all men and .5% of all women are color-blind. A person is chosen at random and found to be color-blind. What is the probability that the person is male? (You may assume that
1. A mouse is put into a T-maze (a maze shaped like a T). In this maze, it has the choice of turning to the left and being rewarded with cheese or going to the right and receiving a mild shock. Before any conditioning takes place (i.e., on trial 1), the mouse is equally likely to go to the left or
A bag is equally likely to contain either one white ball or one red ball. A white ball is added to the bag, and then a ball is selected at random from the bag. If the selected ball is white, what is the probability that the bag originally contained a white ball? (This problem has been attributed to
Kim has a strong first serve; whenever it is good (that is, in), she wins the point 75% of the time. Whenever her second serve is good, she wins the point 50% of the time. Sixty percent of her first serves and 75% of her second serves are good. (a) What is the probability that Kim wins the point
1. When a tennis player hits his first serve as hard as possible (called a blast), he gets the ball in (that is, within bounds) 60% of the time. When the blast first serve is in, he wins the point 80% of the time. When the first serve is out, his gentler second serve wins the point 45% of the time.
1. Refer to Exercise 23. What is the probability that there will be an accidental nuclear war during the next n years? Refer to Exercise 23, Suppose that, during any year, the probability of an accidental nuclear war is .0001 (provided, of course, that there hasn't been one in a previous year).
1. Suppose that, instead of tossing a coin, the player in Exercise 25 draws up to five cards from a deck consisting only of three red and three black cards. The player wins as soon as the number of red cards exceeds the number of black cards and loses as soon as three black cards have been drawn.
1. Refer to Exercise 27. Suppose that a batch of 99 pea plants contains 33 plants of each of the three genotypes. Refer to Exercise 27, Traits passed from generation to generation are carried by genes. For a certain type of pea plant, the color of the flower produced by the plant (either red or
1. A light-bulb manufacturer knows that .05% of all bulbs manufactured are defective. A testing machine is 99% effective; that is, 99% of good bulbs will be declared fine and 99% of flawed bulbs will be declared defective. If a randomly selected light-bulb is tested and found to be defective, what
1. An urn contains five red balls and three green balls. One ball is selected at random and then replaced by a ball of the other color. Then a second ball is selected at random. What is the probability that the second ball is green? 1. Two people toss two coins each. What is the probability that
1. An urn contains four red marbles and three green marbles. One marble is removed, its color noted, and the marble is not replaced. A second marble is removed and its color noted. (a) What is the probability that both marbles are red? Green? (b) What is the probability that exactly one marble is
Bud is a very consistent golfer. On par-three holes, he always scores a 4. Lou, on the other hand, is quite erratic. On par-three holes, Lou scores a 3 seventy percent of the time and scores a 6 thirty percent of the time. (a) If Bud and Lou play a single par-three hole together, who is more likely
1. Consider three dice: one red, one blue, and one green. The sides of the red die contain the numbers 3 3 3 3 3 6, the sides of the blue die contain the numbers 2 2 2 5 5 5, and the sides of the green die contain the numbers 1 4 4 4 4 4.(a) Determine the probability that the red die will show a
Apply to medical diagnostic tests. 1. (True or False) Sensitivity also can be called the true positive rate. 2. (True or False) Specificity also can be called the true negative rate. 3. (True or False) Specificity = 1 - false negative rate. 4. (True or False) Sensitivity = 1 - false positive
1. Suppose that a test for hepatitis has a sensitivity of 95% and a specificity of 90%. A person is selected at random from a large population, of which .05% of the people have hepatitis, and given the test. What is the positive predictive value of the test?2. The probability .0002 (or .02%) in

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