PROBABILITY EXERCISES, Summaries of Business Accounting

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PROBABILITY

EXERCISES

Prof. Dr. Onur KÖKSOY

1. Suppose a family contains two children of different ages, and we are interested in the gender of these children. Let F denote that a child is female and M that the child is male and let a pair such as FM denote that the older child is female and the younger is male. There are four points in the set S of possible observations:

S = {FF, FM, MF, MM}.

Let A denote the subset of possibilities containing no males; B, the subset containing two males; and C, the subset containing at least one male. List the elements of ̅

Solution:

2. Draw Venn diagrams to verify DeMorgan’s laws. That is, for any two sets A and B, ̅̅̅̅̅̅̅̅̅̅̅ = ̅∩ ̅ and ̅̅̅̅̅̅̅̅̅̅̅ = ̅ ̅.

Solution:

5. A manufacturer has five seemingly identical computer terminals available for shipping. Unknown to her, two of the five are defective. A particular order calls for two of the terminals and is filled by randomly selecting two of the five that are available. a. List the sample space for this experiment. b. Let A denote the event that the order is filled with two nondefective terminals. List the sample points in A. c. Construct a Venn diagram for the experiment that illustrates event A. d. Assign probabilities to the simple events in such a way that the information about the experiment is used and the axioms in Definition 2.6 are met. e. Find the probability of event A.

Solution:

a. Let the two defective terminals be labeled D1 and D2 and let the three good terminals be labeled G1, G2, and G3. Any single sample point will consist of a list of the two terminals selected for shipment. The simple events may be denoted by E1 = {D1, D2}, E5 = {D2, G1}, E8 = {G1, G2}, E10 = {G2, G3}. E2 = {D1, G1}, E6 = {D2, G2}, E9 = {G1, G3}, E3 = {D1, G2}, E7 = {D2, G3}, E4 = {D1, G3}, b. Thus, there are ten sample points in S, and. Event A = {E8, E9, E10}. c.

d. Because the terminals are selected at random, any pair of terminals is as likely to be selected as any other pair. Thus, , for , is a reasonable assignment of probabilities.

e. Because ,.

6. Every person’s blood type is A, B, AB, or O. In addition, each individual either has the Rhesus (Rh) factor (+) or does not (−). A medical technician records a person’s blood type and Rh factor. List the sample space for this experiment.

Solution:

7. A sample space consists of five simple events, E1, E2, E3, E4, and E5. a. If , find the probabilities of E4 and E5. b. If , find the probabilities of the remaining simple events if you know that the remaining simple events are equally probable.

Solution:

8. Americans can be quite suspicious, especially when it comes to government conspiracies. On the question of whether the U.S. Air Force has withheld proof of the existence of intelligent life on other planets, the proportions of Americans with varying opinions are given in the table.

Suppose that one American is selected and his or her opinion is recorded.

a. What are the simple events for this experiment? b. Are the simple events that you gave in part (a) all equally likely? If not, what are the probabilities that should be assigned to each?

Solution:

11. A business office orders paper supplies from one of three vendors, V1, V2, or V3. Orders are to be placed on two successive days, one order per day. Thus, (V2, V3) might denote that vendor V2 gets the order on the first day and vendor V3 gets the order on the second day. a. List the sample points in this experiment of ordering paper on two successive days. b. Assume the vendors are selected at random each day and assign a probability to each sample point. c. Let A denote the event that the same vendor gets both orders and B the event that V gets at least one order. Find and by summing the probabilities of the sample points in these events.

Solution:

12. Consider the problem of selecting two applicants for a job out of a group of five and imagine that the applicants vary in competence, 1 being the best, 2 second best, and so on, for 3, 4, and 5. These ratings are of course unknown to the employer. Define two events A and B as:

A : The employer selects the best and one of the two poorest applicants (applicants 1 and 4 or 1 and 5).

B : The employer selects at least one of the two best. Find the probabilities of these events.

Solution :

The experiment involves randomly selecting two applicants out of five. Denote the selection of applicants 3 and 5 by {3 , 5}. 2. The ten simple events, with { i, j } denoting the selection of applicants i and j , are

E 1 : {1 , 2}, E 5 : {2 , 3}, E 8 : {3 , 4}, E 10 : {4 , 5}. E 2 : {1 , 3}, E 6 : {2 , 4}, E 9 : {3 , 5}, E 3 : {1 , 4}, E 7 : {2 , 5}, E 4 : {1 , 5},

A random selection of two out of five gives each pair an equal chance for selection. Hence, we will assign each sample point the probability 1 / 10. That is,

Checking the sample points, we see that B occurs whenever occurs. Hence, these sample points are included in B. Finally, P(B) is equal to the sum of the probabilities of the sample points in B , or

Similarly, we see that event

13. A balanced coin is tossed three times. Calculate the probability that exactly two of the three tosses result in heads.

Solution:

The experiment consists of observing the outcomes (heads or tails) for each of three tosses of a coin. A simple event for this experiment can be symbolized by a three-letter sequence of H’s and T ’s, representing heads and tails, respectively. The first letter in the sequence represents the observation on the first coin. The second letter represents the observation on the second coin, and so on.The eight simple events in S are

16. Two additional jurors are needed to complete a jury for a criminal trial. There are six prospective jurors, two women and four men. Two jurors are randomly selected from the six available. a. Define the experiment and describe one sample point. Assume that you need describe only the two jurors chosen and not the order in which they were selected. b. List the sample space associated with this experiment. c. What is the probability that both of the jurors selected are women?

Solution:

17. The Bureau of the Census reports that the median family income for all families in the United States during the year 2003 was $43,318. That is, half of all American families had incomes exceeding this amount, and half had incomes equal to or below this amount. Suppose that four families are surveyed and that each one reveals whether its income exceeded $43,318 in 2003. a. List the points in the sample space. b. Identify the simple events in each of the following events: A: At least two had incomes exceeding $43,318. B: Exactly two had incomes exceeding $43,318. C: Exactly one had income less than or equal to $43,318. c. Make use of the given interpretation for the median to assign probabilities to the simple events and find P(A), P(B), and P(C).

Solution:

18. A boxcar contains six complex electronic systems. Two of the six are to be randomly selected for thorough testing and then classified as defective or not defective. a. If two of the six systems are actually defective, find the probability that at least one of the two systems tested will be defective. Find the probability that both are defective. b. If four of the six systems are actually defective, find the probabilities indicated in part (a).

Solution:

19. The names of 3 employees are to be randomly drawn, without replacement, from a bowl containing the names of 30 employees of a small company. The person whose name is drawn first receives $100, and the individuals whose names are drawn second and third receive $50 and $25, respectively. How many sample points are associated with this experiment?

By a random assignment of laborers to the jobs, we mean that each of the N sample points has probability equal to 1/N. If A denotes the event of interest and na the number of sample points in A, the sum of the probabilities of the sample points in A is

. The number of sample points in A, , is the number of ways of assigning laborers to the four jobs with the 4 members of the ethnic group all going to job. The remaining 16 laborers need to be assigned to the remaining jobs. Because there remain two openings for job 1, this can be done in

ways. It follows that

Thus, if laborers are randomly assigned to jobs, the probability that the 4 members of the ethnic group all go to the undesirable job is very small. There is reason to doubt that the jobs were randomly assigned.

22. An airline has six flights from New York to California and seven flights from California to Hawaii per day. If the flights are to be made on separate days, how many different flight arrangements can the airline offer from New York to Hawaii?

Solution:

23. A businesswoman in Philadelphia 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 she plans her route. a. How many different itineraries (and trip costs) are possible? b. If the businesswoman randomly selects one of the possible itineraries and Denver and San Francisco are two of the cities that she plans to visit, what is the probability that she will visit Denver before San Francisco?

Solution:

24. An experiment consists of tossing a pair of dice. a. Use the combinatorial theorems to determine the number of sample points in the sample space S. b. Find the probability that the sum of the numbers appearing on the dice is equal to 7.

Solution:

25. How many different seven-digit telephone numbers can be formed if the first digit cannot be zero?

Solution:

28. A local fraternity is conducting a raffle where 50 tickets are to be sold—one per customer. There are three prizes to be awarded. If the four organizers of the raffle each buy one ticket, what is the probability that the four organizers win a. all of the prizes? b. exactly two of the prizes? c. exactly one of the prizes? d. none of the prizes?

Solution :

29. Five firms, F1, F2,... , F5, each offer bids on three separate contracts, C1,C2, and C3. Any one firm will be awarded at most one contract. The contracts are quite different, so an assignment of C1 to F1, say, is to be distinguished from an assignment of C2 to F1. a. How many sample points are there altogether in this experiment involving assignment of contracts to the firms? (No need to list them all.) b. Under the assumption of equally likely sample points, find the probability that F3 is awarded a contract.

Solution:

30. A study is to be conducted in a hospital to determine the attitudes of nurses toward various administrative procedures. A sample of 10 nurses is to be selected from a total of the 90 nurses employed by the hospital. a. How many different samples of 10 nurses can be selected? b. Twenty of the 90 nurses are male. If 10 nurses are randomly selected from those employed by the hospital, what is the probability that the sample of ten will include exactly 4 male (and 6 female) nurses?

Solution:

31. Two cards are drawn from a standard 52-card playing deck. What is the probability that the draw will yield an ace and a face card?

Solution:

32. Five cards are dealt from a standard 52-card deck. What is the probability that we draw a. 1 ace, 1 two, 1 three, 1 four, and 1 five (this is one way to get a “straight”)? b. any straight?

35. Consider the following events in the toss of a single die: A: Observe an odd number, B: Observe an even number, C: Observe a 1 or 2. a. Are A and B independent events? b. Are A and C independent events?

Solution

a. To decide whether A and B are independent, we must see whether they satisfy the conditions of Definition 2.10. In this example, P(A) = 1/2, P(B) = 1/2, and P(C) = 1/3. Because A ∩ B = ∅, P(A|B) = 0, and it is clear that P(A|B) = P(A). Events A and B are dependent events. b. Are A and C independent? Note that P(A|C) = 1/2 and, as before, P(A) = 1/2. Therefore, P(A|C) = P(A), and A and C are independent.

36. Three brands of coffee, X, Y , and Z, are to be ranked according to taste by a judge. Define the following events:

A: Brand X is preferred to Y. B: Brand X is ranked best. C: Brand X is ranked second best. D: Brand X is ranked third best.

If the judge actually has no taste preference and randomly assigns ranks to the brands, is event A independent of events B, C, and D?

Solution:

The six equally likely sample points for this experiment are given by

where XY Z denotes that X is ranked best, Y is second best, and Z is last. Then and it follows that

Thus, events A and C are independent, but events A and B are dependent. Events A and D are also dependent.

37. If two events, A and B, are such that , find the following: a. P(A|B) b. P(B|A) c. P(A|A B) d. P(A|A ∩ B) e. P(A ∩ B|A B)

Solution:

38. Gregor Mendel was a monk who, in 1865, suggested a theory of inheritance based on the science of genetics. He identified heterozygous individuals for flower color that had two alleles (one r = recessive white color allele and one R = dominant red color allele). When these individuals were mated, 3/4 of the offspring were observed to have red flowers, and 1/4 had white flowers. The following table summarizes this mating; each parent gives one of its alleles to form the gene of the offspring.

We assume that each parent is equally likely to give either of the two alleles and that, if either one or two of the alleles in a pair is dominant (R), the offspring will have red flowers. What is the probability that an offspring has

a. at least one dominant allele? b. at least one recessive allele? c. one recessive allele, given that the offspring has red flowers?