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introduction to probability statistics

Introduction To Probability And Statistics 3rd Edition William Mendenhall - Solutions

Let A, B, and C be three events in a sample space. If A ⊆ B and the events A and C are disjoint, then the events B and C are also disjoint.
Let A be an event in a sample space such that P(A) = 1, and B be another event on that space. Then, P(A ∪ B′) = 0.
For any events A and B in a sample space, we have (A ∪ B)′ = A′B′.
If an event A satisfies the condition P(A) = 1 − 3P(A′), then P(A) = 1∕4.
If A, B are disjoint events in a sample space and we know that A occurs, then B does not occur.
The sample space for an experiment is the closed interval [0, 4] of the real line.Then, Ω is a finite sample space.
For two events A and B in a sample space, the probability that at least one of them occurs is P(A ∪ B).
When throwing a die, the event “the outcome is a multiple of 3” is an elementary outcome.
We play the following game 50 times. We throw a die and, if the outcome is 1, 2, 3 or 4 we win, otherwise we lose:A: we win at least 40 times;B: we win at most 25 times.
Each of five boxes contain 15 red balls and 25 black balls. We select a ball from the first box randomly and place it into the second one. Then, we select a ball from the second box and place it into the third one, and so on, until finally we select a ball from the 5th box:A: the ball selected from
Each of two boxes contain 20 red balls and 30 black balls. We select a ball from the first box randomly and place it into the second one. Then, we select a ball from the second box and place it into the first one.A: both balls selected were red;B: the ball selected from the second box was black;C:
We throw a die until we get three consecutive identical outcomes (e.g. 111, 222 and so on).A: the experiment finishes after more than 20 throws;B: the experiment terminates at the 12th throw;C: the experiment finishes at or before the 20th throw.
We throw a die until a six appears for the first time:A: the experiment is completed after exactly 3 throws;B: the experiment is completed after at least 3 throws;C: the experiment is completed after at most 3 throws.
We roll a die twice:A: the first outcome is 5 and the second one is greater than 2;B: the first outcome is at least 4 and the second outcome is an even integer;C: the sum of the two outcomes is 5;D: the sum of the two outcomes is at least 9;E: the difference between the two outcomes is at most 3.
We toss a coin four times:A: the first and the third outcomes are “Heads”;B: exactly two heads appear;C: the number of heads is 3 or more.
(Borel–Cantelli lemma). Let {An} be a sequence of events in a sample space Ω, such that the sequence of real numbersconverges to a real number. Prove thatthat is, the probability that infinitely many, among the events A1, A2,…, occur is zero.(Hint: Apply Proposition 1.11 to the sequence of
Let {An} be a decreasing sequence of events on a sample space, for which we have P(A1) = q, P(An − An+1) = qnp, n = 1, 2,…, where 0 < q < 1 and p = 1 − q.(i) Show that P(An) = qn for each n = 1, 2,….(ii) Calculate the probability P(A) for the limiting event A = lim n→∞An.
Let the sample space Ω for an experiment be the set of real numbers. In this space, we define the eventsAssuming that for n = 1, 2,…, the probability of the event An iscalculate the probability of the open interval (2, 4). An = + n = 1, 2,.... 2"
Let the sample space Ω for an experiment be the set of real numbers, a be a real number, and let further the probabilities of the events (in this case, the open intervals)be given byCalculate the probability of the elementary event {a}. A = (a-a+), n = 1, 2,...,
This exercise shows that an arbitrary sequence of events in a sample space can be written as the union of increasing events. Let {An}n≥1 be a sequence of events in a sample space (not necessarily monotone), and define a new sequence {Cn}n≥1 by Cn = ∪n i=1Ai, n = 1, 2,….(i) Show that
Let the sample space for an experiment be the set of positive integers, and consider the sequence of events {An}n≥1 defined in it, where An is given byShow that {An}n≥1 is a monotone sequence and find its limit as n → ∞. A = {n,n+1,n+2,...), for n = 1, 2,....
For the events A1, A2, and A3 in the same sample space, it is known that P(Ai) = 1 2i , i = 1, 2, 3, and P(AiAi+1) = 1 2i+2 , i = 1, 2.If, in addition, we know that the events A1 and A3 are mutually exclusive, calculate the probability that at least one of A1, A2, A3 occur.
Consider the events A1, A2,…, An defined in a sample space Ω. Show that the probability that none of them appear is equal towhere S1, S2,…, Sn are the sums defined earlier in Proposition 1.10, and S0 = 1.Application: For n = 3 assume that P(Ai) = 1∕3 for i = 1, 2, 3. Moreover, assume that
For the events A, B, and C of a sample space, we are given P(ABC) =a, P(A′BC) = b1, P(AB′C) = b2, P(ABC′) = b3, P(AB′C′) = c1, P(A′BC′) = c2, P(A′B′C) = c3.Calculate the probabilities of the events A, B, and C in terms ofa, bi, and ci.Similarly find for the events A − B, B −
Let A, B, and C be three events such that C ⊆ B ⊆ A. Let a = P(A), b = P(B), and c = P(C).(i) Calculate the probabilities P(A − B), P(A − (B − C)), P((A − B) − C)in terms ofa, b,c. What do you observe?(ii) Obtain a numerical answer to the result in (i) for a = 1∕2, b = 1∕4, c =
Suppose A1, A2, and A3 are three events in a sample space. Let us consider the sums S1 = P(A1) + P(A2) + P(A3), S2 = P(A1A2) + P(A1A3) + P(A2A3), S3 = P(A1A2A3).Express mathematically, in terms of S1, S2, and S3, the probability for the event that(i) exactly one of the Ai occurs;(ii) exactly two of
With reference to the out-patient visits to a hospital, we consider the following events:A: a patient is over 50 years of age;B: a patient is under 25 years of age;C: the condition of a patient upon arrival is considered to be serious;D: a patient needs to stay at the hospital for at least one
Let A1, A2, and A3 be three events in a sample space Ω, which are not necessarily pairwise disjoint.(i) Prove that P(A1 ∪ A2 ∪ A3) ≤ P(A1) + P(A2) + P(A3).(ii) Show that the equality P(A1 ∪ A2 ∪ A3) = P(A1) + P(A2) + P(A3)holds if and only if P(A1A2) + P(A1A3) + P(A2A3) = P(A1A2A3).
Let A and B be two events in a sample space Ω such that P(A) = 0 and P(B) = 0.Use the results of Exercise 5 to show that P(A ∪ B) = 0, P(AB) = 0, P(A′B′) = 1.
Let A and B be two events in a sample space Ω such that P(A) = 1 and P(B) = 1.Use the results of Exercise 5 to show that P(A ∪ B) = 1, P(AB) = 1, P(A′B′) = 0.
Let A and B be two events in a sample space Ω with P(A) = ???? and P(B) = ????. Confirm the validity for each of the following:(i) P(A′B′) = 1 − ???? − ???? + P(AB);(ii) ???? + ???? − 1 ≤ P(AB) ≤ min{????, ????};(iii) max{????, ????} ≤ P(A ∪ B) ≤ ???? + ????.Suppose P(A) = 0.75
Suppose for the events A and B we have 2P(A) = 3P(B) = 4P(AB) and P(A′B) = 0.05.(i) Calculate the probabilities P(A), P(B), and P(AB).(ii) Find the probabilities of the following events A ∪ B, A′B′, AB′, AB′ ∪ A′B, A ∪ B′, A′ ∪ B.
Let A and B be two events in a sample space Ω. Establish the relationship P(A′)P(B) − P(A′B) = P(A)P(B′) − P(AB′) = P(A′B′) − P(A′)P(B′)= P(AB) − P(A)P(B).
When Sandra drives back home from work, she has to pass through two sets of traffic lights. The probability that she has to stop at the first is 0.35, while the probability that she has to stop at the second one is 0.40. Finally, the probability that she does not stop at a traffic light is 0.50.
For a specific area, the probability that it is hit by a typhoon during a year is 0.02. For the same period, the probability that the area suffers severe damage due to excessive rain is 0.05, while the probability that both these events occur in a year is 0.001. Find the probability that over a
Consider the continuous sample space Ω = [1, 1000]. For each event A in Ω(A ⊆ Ω), we define the probability P(A) to be k∕1000, where k is the number of integers included in the set A. Then:(i) Show that P(⋅) satisfies the three properties of Definition 1.10.(ii) Calculate the probabilities
For two disjoint events A and B in a sample space, suppose P(A ∪ B) = 1 2, 3P(A′) + 2P(B) = 3.Then, find the probabilities P(A) and P(B).
Let A1, A2, A3, and A4 be four events in a sample space Ω that are pairwise disjoint and such thatIf it is known that P(A2) = 2P(A1), P(A4) = 3P(A2), P(A2) = 4P(A3), calculate the probabilities of the union of any two and any three events among A1, A2, A3, A4. 4 . UA =Q2. i=1
Consider the infinite countable sample space Ω = {0, 1, 2,…}. On this space, assume that the probabilities of the elementary events {i}, for i = 0, 1,…, are given by(i) Show that the probability that the outcome of the experiment is at least k, for k = 1, 2,…, is given by (3∕4)k.(ii) What
Let A, B, and C be three events in a sample space Ω such that AB = ∅. Then:(i) Verify that the events AC and BC are disjoint.(ii) Calculate the probability of the event(A′ ∪ C′)(B′ ∪ C′)in terms of the probabilities P(AB) and P(AC).
Let A and B be two disjoint events in a sample space Ω. Prove that P(A′B′) = P(A′) + P(B′) − 1.
Let A1, A2,…, An be a finite collection of events on a sample space Ω such that A1 ∪ A2 ∪ · · · ∪ An = Ω and the Ai’s are pairwise disjoint. If it is known that P(Ai) = 3P(Ai+1) for i = 1, 2,…, n − 1, find P(A3) and P(A5) (assuming n ≥ 5).
When a salesman visits a certain town, he stays in one of three available hotels, H1, H2, H3. Let Ai be the event that he stays in Hotel Hi, for i = 1, 2, 3. If it is known thatfind the probability that he stays in Hotel Hi, for i = 1, 2, 3. P(A3) P(A UA) 2P(A3) and 3P(A) = 1- 2
The probability that the price of a certain stock increases during a day is 50% higher than the probability that the price of the stock decreases, while it is also three times as much as the probability that the price remains the same. Find the probability that, during a day, the price of the stock
At 08:00 a.m., the probability that John is in bed is 0.2, while the probability of him having breakfast is 0.5. What is the probability that on a particular day he is neither in bed nor having breakfast at 08:00 a.m.?
If for the event A we have 2P(A) = P(A′) + 0.5, find the probability P(A).
When Jenny returns home from her local supermarket, she has to pass through three sets of traffic lights. During the last 50 times she did that, she had to stop her car at the first set of traffic lights on 30 occasions, while 10 times the first traffic light was green but she had to stop at the
In a University class, there are 200 students, who at the end of the last semester took three exams: Algebra (A), Calculus (C), and Probability (P). The numbers of students who obtained a first class mark were 24, 12, and 33 for A,C, and P, respectively.The numbers of students who obtained a first
In a survey conducted to examine whether there is correlation between gender and physical exercise, 700 persons were selected at random and their gender and whether they did some form of physical exercise at least twice a week or not were recorded.The outcomes are given in the following
In a study of religious habits of persons living in a city, 500 persons were asked whether they go to church regularly. 240 of them were men, out of which 40 said that they attend a church service regularly, while among women, 80 said that they attend a church service regularly. Consider the
For the experiment of throwing a die twice, use the results from Table 1.1 to calculate the probabilities of the following events:D: at least one of the two outcomes is 3;E: the outcome of the second throw is 5;F: the results of the two throws are the same.How do the probabilities of the events D
Use the first 50 throws from the data in Table 1.1 to calculate the probabilities of the events A, B, and C in Example 1.8. Then, repeat the same calculations using the second half of the throws (numbered 51–100 on the table). What do you observe?
For any events A and B in the same sample space Ω, we define the symmetric difference of A and B to be the event (see next figure)(i) Express in words what this event represents;(ii) Show that the following hold:AΔB = BΔA, AΔB = (A ∪ B) − AB, (AΔB)ΔB = A. = AAB (AB)U (B-A). A B 02
For any events A, B, and C in a sample space, verify the truth of the following relations:(i) (A − B) − C = (A − C) − (B − C);(ii) (B − A) ∪ (C − A) = (B ∪ C) − A;(iii) A − (B − C) = (A − B) ∪ (AC);(iv) (A − B) ∪ B = A ∪ B;(v) A(B − C) = AB − AC;(vi) A ∪ B
Let A, B, and C be three events in a sample space Ω. In each case below, find an event X such that the union operator on the right hand side is applied between events which are disjoint (for example, in (i) we seek X such that AB and X are disjoint sets):(i) B = (AB) ∪ X;(ii) A ∪ B = (A′B)
From an ordinary card deck with 52 cards, we select n cards successively. We consider the events A1: the first card drawn is an ace;An: in the first n − 1 selections (n ≥ 2), neither an ace nor a face card is drawn and in the nth selection an ace is drawn, where n = 2, 3,… Write down an
Simplify each of the expressions below by the use of properties among event operators:(i) A′B′AB;(ii) (A′ ∪ B′) ∪ AB;(iii) (A ∪ B)(A ∪ B′)(A′ ∪ B);(iv) (A′B′)(A ∪ B);(v) [(A ∪ B)(A ∪ B′)] ∪ [(A′ ∪ B)(A′ ∪ B′)];(vi) (A ∪ B ∪ C)(A ∪ B ∪ C′)(A
Let Ai, i = 1, 2,…, n, be events on the same sample space Ω. Express in words what conclusions can be drawn about these events in each of the following cases:(i) A1 ∪ A2 ∪ · · · ∪ An = A1;(ii) A1A2 · · · An = A1;(iii) A1 ∪ A2 ∪ · · · ∪ An = A1 ∪ An;(iv) A1A2 · · · An =
In order to describe a chance experiment, we have used the following (continuous)sample spaceΩ = {(x, y) ∶ −5 ≤ x ≤ 5 and − 3 ≤ y ≤ 7}.On this space, we define the following events:A = {(x, y) ∈ Ω ∶ x = y}, B = {(x, y) ∈ Ω ∶ x2 = y2}, C = {(x, y) ∈ Ω ∶ x + y = 0}, D =
Express each of the following events in terms of the events Ai, Bi,C,D defined in Exercise 9.(i) The number on the ball selected is greater than 4 and less than 10.(ii) The ball selected is either white or has a number greater than 10 on it.(iii) The ball selected is black or has an even number on
A box contains 15 balls numbered 1, 2,…, 15. The balls numbered 1–5 are white and those numbered 6–15 are black. We select a ball at random, and record its color and the number on it.(i) Write down a suitable sample space for this experiment.(ii) We define the events Ai: the ball selected has
Let A, B, and C be three events on a sample space Ω. Examine, possibly with the aid of Venn diagrams, which of the following results are always true:(i) (A − AB)B = AB;(ii) (A ∪ B)′C = A′B′C;(iii) (AB)′C′ = (ABC)′;(iv) AB ⊆ A ⊆ A ∪ B ∪ C;(v) A′B ∪ C′B = (A ∪
Suppose the events A and B of a sample space Ω are such that A ⊆ B and A′ ⊆ B.Then prove that B = Ω. (Hint: You may use the result of Exercise 5.)
If, for the events A and B, we know that A ⊆ B and A ⊆ B′, show that A = ∅. (Hint:Use the result of Exercise 5.)
Suppose for the events A, B,C,D, we have A ⊆ B and C ⊆ D. Then, arguing as in the proof of Proposition 1.2, show that A ∪ C ⊆ B ∪ D and AC ⊆ BD.
What conclusions can we draw about the events A and B if the following relations hold?(i) A ∪ B = A; (ii) A − B = A;(iii) AB = A; (iv) A − B = B − A.
For each of the following graphs (a)–(d), express the event in the shaded area in terms of the events A, B, and C and state, in words, what this event represents. A A (a) B C (c) C B A B A (b) B C (d)
Consider the experiment of throwing a die twice, and define the following events:A: the sum of the two outcomes is 6;B: the two outcomes are equal;C: the first outcome is an even integer;D: the first outcome is an odd integer.Write down the 36 sample elements for this experiment, and then identify
Let A, B, and C be three events in a sample space Ω. Express each of the following events by the use of the operators (unions, intersections, complements) among sets:(i) all three events occur;(ii) at least one of the three events occur;(iii) A occurs, but not A and B;(iv) A and C occur, but not
In a water supply network, depicted below, the water is transferred from point A to point F through water tubes. At the positions marked with the numbers 1, 2, 3, and 4 on the graph, there are four switches which, if turned off, stop the water supply passing through the tube.(i) Find a sample space
At a car production line in a factory, each engine produced is tested to examine whether it is ready for use or has some fault. If two consecutive engines that are examined are found faulty, the production process stops and is revised (in such a case, the experiment is terminated). Otherwise, the
A bus, which has a capacity of carrying 50 passengers, passes through a certain bus stop every day at some time point between 10:00 a.m. and 10:30 a.m. In order to study the time the bus arrives at this stop, as well as the number of passengers in the bus at the time of its arrival, we use the
Bill just visited a friend who lives in Place A of the graph below and he wants to return home, which is at Place I on the graph. In order to minimize the distance he has to walk, he moves either downwards (e.g. from Place A to Place D) or to the right (e.g. from Place E to Place F on the graph).
Mary has in her wallet three $1 coins, one $2 coin and four coins of 25 ¢. She selects four coins at random from her wallet.(i) Write down a sample space for the possible selections she can make.(ii) Express the following events as subsets of the above sample space:C1: exactly three 25 ¢ coins
Irène has four books that she wants to put on a library shelf. Three of these books form a 3-volume set of a dictionary, so that they are marked as Volumes I, II, and III, respectively.(i) Find an appropriate sample space for all possible ways she can put the books on the shelf.(ii) Identify the
A box contains 3 red balls and 2 yellow balls. Give a suitable sample space to describe all possible outcomes for the experiment of selecting 4 balls at random, in each of the following schemes:(i) For each ball selected, we note its color and return it to the box so that it is available for the
A company salesman wants to visit the four citiesa, b,c, d wherein his company has stores. If he plans to visit each city once, give a suitable sample space to describe the order in which he visits the cities. Then identify, by an appropriate listing of its elements, each of the following
We throw a die twice. Give a suitable sample space for this experiment and then identify the elements each of the following events contains:A1: the outcome of the first throw is 6;A2: the outcome of the second throw is a multiple of 3;A3: the outcome of the first throw is 6 and the outcome of the
A digital scale has an accuracy of two decimal places shown on its screen. Each time a person steps on the scale, we record his/her weight by rounding it to the closest integer (in kilograms). Thus, if the decimal part is 0.50 or greater, we round up to the next integer.(i) Give an appropriate
We toss a coin until either Heads appear for the first time or we have five tosses which all result in Tails. Give a suitable sample space for this experiment, and then write explicitly (i.e. by listing their elements) each of the following events:A: the experiment terminates at the third toss of
John throws a die and subsequently he tosses a coin.(i) Suggest a suitable sample space that describes the outcomes of this experiment.(ii) Let A be the event that “the outcome of the coin toss is Heads.” Which elements of the sample space are included in the event A?
Provide suitable sample spaces for each of the following experiments. For each sample space, specify whether it is finite, infinitely countable or uncountable.(a) Two throws of a die(b) Gender of the children in a family that has three children(c) Random selection of a natural number less than
An occupational hygienist believes that a two-hour training session on proper hand washing will improve time spent on hand washing. Based on a random sample of 25 high school students who had attended the session, their hand washing times (in seconds) were recorded as follows:a. Calculate the
Have goals been easier to score in the NHL in some eras than others? Many of us have heard of hockey greats such as Maurice "Rocket" Richard, Gordie Howe, Wayne Gretzky, and Mario Lemieux. But have you heard of Bill Cook, who scored 28 goals in 48 games for the New York Rangers in 1933, or Norm
2.82 Arranging Objects, continued Refer to Exercise 2.81.a. Find the five-number summary for this data set.b. Construct a box plot for the data.c. Are there any unusually large or small response times identified by the box plot?d. Construct a stem and leaf display for the response times. How would
2.81 Arranging Objects The following data are the response times in seconds for n = 25 first graders to arrange three objects by size.a. Find the mean and the standard deviation for these 25 response times.b. Order the data from smallest to largest.c. Find the z-scores for the smallest and largest
2.80 Breathing Patterns Research psychologists are interested in finding out whether a person's breathing patterns are affected by a particular experimental treatment. To determine the general respiratory patterns of the n = 30 people in the study, the researchers collected some baseline
2.79 Environmental Factors How do Canadians rate environmental factors in terms of the threat they pose to Canada? Below are findings of the survey conducted by the Strategic Counsel.22 The large majority of Canadians (61%) believe that toxic chemicals are linked to human health. 55% think air
2.78 Ages of Pennies Here are the ages of 50 pennies from Exercise 1.45 and data set EX0145. The data have been sorted from smallest to largest.a. What is the average age of the pennies?b. What is the median age of the pennies?c. Based on the results of parts a andb, how would you describe the age
2.77 Bobby Hull Two box plots of Bobby Hull's goal scores are given below.21 One is for 1957-1975, and the other includes the years 1975-1980.The statistics used to construct these box plots are given in the table.a. Calculate the upper fences for both of these box plots.b. Can you explain why the
2.76 Great Goal Scorers The number of goals scored per season by each of four NHL superstars over each player's career were recorded and shown in the box plots below.Write a short paragraph comparing the goal scoring patterns of these four players. Mario Lemieux H Waynetzky Bobby Hull 0 20 20 40
2.75 Is It Accurate? From the following data, a student calculated s to be 0.263. On what grounds might we doubt his accuracy? What is the correct value (to the nearest hundredth)? 17.2 17.1 17.0 17.1 16.9 17.0 17.1 17.0 17.3 17.2 17.1 17.0 17.1 16.9 17.0 17.1 17.3 17.2 17.4 17.1
2.74 University Professors Consider a population consisting of the number of professors per university at small universities. Suppose that the number of professors per university has an average = 175 and a standard deviation or = 15.a. Use Tchebysheff's Theorem to make a statement about the
2.73 Parasites in Foxes A random sample of 100 foxes was examined by a team of veterinarians to determine the prevalence of a particular type of parasite. Counting the number of parasites per fox, the veterinarians found that 69 foxes had no parasites, 17 had one parasite, and so on. A frequency
2.72 TV Commercials The mean duration of television commercials on a given network is 75 seconds, with a standard deviation of 20 seconds. Assume that durations are approximately normally distributed.a. What is the approximate probability that a commercial will last less than 35 seconds?b. What is
2.71 Social Ambivalence The following data represent the social ambivalence scores for 15 people as measured by a psychological test. (The higher the score, the stronger the ambivalence.)a. Guess the value of s using the range approximation.b. Calculate x and s for the 15 social ambivalence
2.70 Lumber Rights A company interested in lumbering rights for a certain tract of slash pine trees is told that the mean diameter of these trees is 35 cm with a standard deviation of 7 cm. Assume the distri- bution of diameters is roughly mound-shaped.a. What fraction of the trees will have
2.69 Drugs for Hypertension A pharmaceutical company wishes to know whether an experimental drug being tested in its laboratories has any effect on systolic blood pressure. Fifteen randomly selected subjects were given the drug, and their systolic blood pressures (in millimetres of mercury) are
2.68 Long-Stemmed Roses A strain of long-stemmed roses has an approximate normal distribution with a mean stem length of 38 cm and standard deviation of 5.5 cm.a. If one accepts as "long-stemmed roses" only those roses with a stem length greater than 38 cm, what percentage of such roses would be
2.67 Standardized Test Scores A district's school board's verbal and mathematics scholastic aptitude tests are scored on a scale of 200 to 800. Although the tests were originally designed to produce mean scores of approximately 500, the mean verbal and math scores in recent years have been as low

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