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introduction to probability statistics

Probability And Statistics For Engineers 9th Global Edition Richard Johnson, Irwin Miller, John Freund - Solutions

With reference to Exercise 3.34 and Figure 3.11, assume that each of 150 persons has the same chance of being selected, and find the probabilities that he or she(a) lives more than 3 miles from the center of the city given that he or she would gladly switch to public mass transportation;(b)
With reference to Figure 3.13, find(a) \(P(A \mid B)\);(b) \(P(B \mid \bar{C})\);(c) \(P(A \cap B \mid C)\);(d) \(P(B \cup C \mid \bar{A})\);(e) \(P(A \mid B \cup C)\);(f) \(P(A \mid B \cap C)\);(g) \(P(A \cap B \cap C \mid B \cap C)\);(h) \(P(A \cap B \cap C \mid B \cup C)\).Data From Figure 3.13
With reference to the used car example and the probabilities given in Figure 3.9, find(a) \(P\left(M_{1} \mid P_{1}\right)\) and compare its value with that of \(P\left(M_{1}\right)\)(b) \(P\left(C_{3} \mid P_{2}\right)\) and compare its value with that of \(P\left(C_{3}\right)\);(c) \(P\left(M_{1}
With reference to Exercise 3.47, find the probabilities that the company will get the tender for constructing an underpass given that(a) it got the tender for constructing a flyover;(b) it did not get the tender for constructing a flyover.Data From Exercise 3.47 3.47 The probability that a
Prove that \(P(A \mid B)=P(A)\) implies that \(P(B \mid A)=\) \(P(B)\) provided that \(P(A) eq 0\) and \(P(B) eq 0\).
In a certain city, sports bikes are being targeted by thieves. Assume that the probability of a sports bike being stolen is 0.09 while the probability is only 0.5 for a regular bike. Taking, as an approximation for all bikes in that area, the nationwide proportion 0.19 of sports bikes, find(a) the
Given that \(P(A)=0.60, P(B)=0.40\), and \(P(A \cap B)=\) 0.24 , verify that(a) \(P(A \mid B)=P(A)\);(b) \(P(A \mid \bar{B})=P(A)\);(c) \(P(B \mid A)=P(B)\);(d) \(P(B \mid \bar{A})=P(B)\).
Among 40 condensers produced by a machine, 6 are defective. If we randomly check 5 condensers, what are the probabilities that(a) none are defective;(b) all are defective?
Among 50 students enrolled in a college, 40 had applied for civil engineering and 10 had applied for mechanical engineering. If two students have been enrolled in software engineering by mistake, and the "selection" is random, what are the probabilities that(a) both had applied for civil
If the odds are 5 to 3 that an eventM will not occur, 2 to 1 that event N will occur, and 4 to 1 that they will not both occur, are the two events M and N independent?
Find the probabilities of getting(a) eight heads in a row with a balanced coin;(b) three 3's and then a 4 or a 5 in four rolls of a balanced die;(c) five multiple-choice questions answered correctly, if for each question the probability of answering it correctly is \(\frac{1}{3}\).
For three or more events which are not independent, the probability that they will all occur is obtained by multiplying the probability that one of the events will occur, times the probability that a second of the events will occur given that the first event has occurred, times the probability that
Use the information on the tree diagram of Figure 3.17 to determine the value of(a) \(P(Y)\);(b) \(P(X \mid Y)\);(c) \(P(X \mid \bar{Y})\). 0.45 X 0.35 Y X 0.75 Y
There are over twenty thousand objects orbiting in space. For a given object, let \(A\) be the event that the charred remains do hit the earth. Suppose experts, using their knowledge of the size and composition of the object as well as its re-entry angle, determine that \(P(A)=0.25\).Next, let
An insurance company's records of 12,299 automobile insurance policies showed that 2073 policy holders made a claim (Courtesy J. Hickman). Among insured drivers under age 25, there were 1032 claims out of 5192 policies. For person selected at random from the policy holders, let \(A=\) [Claim was
Identity theft is a growing problem in the United States. According to a Federal Trade Commission Report about 280,000 identity complaints were filed for 2011. Among the 43.2 million persons in the 20-29 year old age group, 56,689 complaints were filed. The 20-29 year old age group makes up
With reference to the Example 30, for a problem diagnosed as being due to an incomplete initial repair, find the probability that the initial repair was made by(a) Tom;(b) Georgia;(c) Peter.Data From Example 30 EXAMPLE 30 Variances of X - X2 and X + X2 when X, and X2 are independent Let X have mean
Two firms \(V\) and \(W\) consider bidding on a roadbuilding job, which may or may not be awarded depending on the amounts of the bids. Firm \(V\) submits a bid and the probability is 0.8 that it will get the job provided firm \(W\) does not bid. The probability is 0.7 that \(W\) will bid, and if
Engineers in charge of maintaining our nuclear fleet must continually check for corrosion inside the pipes that are part of the cooling systems. The inside con-dition of the pipes cannot be observed directly but a nondestructive test can give an indication of possible corrosion. This test is not
An East Coast manufacturer of printed circuit boards exposes all finished boards to an online automated verification test. During one period, 900 boards were completed and 890 passed the test. The test is not infallible. Of 30 boards intentionally made to have noticeable defects, 25 were detected
(a) Last year, 425 companies applied for 52 tenders floated by the government. This year, you will be applying for one of 52 similar tenders being floated, and would like to estimate the probability of being allotted one. Give your estimate and comment on one factor that might influence your
With reference to the preceding exercise, express each of the following events symbolically by listing its elements, and also express it in words:(a) \(\bar{Y}\);(b) \(X \cap Y\);(c) \(Y \cup Z\);(d) \(X \cup \bar{Y}\).
Use Venn diagrams to verify that(a) \(\bar{A} \cap B=\overline{A \cup \bar{B}}\);(b) \(A \cup B=\overline{\bar{A} \cap \bar{B}}\).
The quality of surround sound from four digital movie systems is to be rated superior, average, or inferior, and we are interested only in how many of the systems get each of these ratings. Draw a tree diagram which shows the 12 different possibilities.
CCTV cameras are to be fitted at six fixed places in a bank. In how many ways can the six available CCTV cameras be fitted at the six places in the bank?
In how many ways can 4 out of 11 similar propellers be fitted on an airplane?
Given \(P(A)=0.30, P(B)=0.40\), and \(P(A \cap B)=\)0.20, find(a) \(P(A \cup B)\);(b) \(P(\bar{A} \cap B)\);(c) \(P(A \cap \bar{B})\);(d) \(P(\bar{A} \cup \bar{B})\).(e) Are \(A\) and \(B\) independent?
In a sample of 652 engines tested, only 28 of them have cylinders with a mild leak. Estimate the probability that an engine tested will have a leak in its cylinder.
The marketing manager reported to the head engineer regarding a survey concerning the company's portable cleaning tool. He claims that, among the 200 customers surveyed, 165 said the product is reliable, 117 said it is easy to use, 88 said it is both reliable and easy to use, and 33 said it is
The probabilities that a satellite launching rocket will explode during lift-off or have its guidance system fail in flight are 0.0002 and 0.0005 . Assuming independence find the probabilities that such a rocket will(a) not explode during lift-off;(b) explode during lift-off or have its guidance
Given \(P(A)=0.40, P(B)=0.55\) and \(P(A \cap B)=\)0.10, find(a) \(P(\bar{A} \mid \bar{B})\);(b) \(P(\bar{A} \mid B)\);(c) \(P(\bar{B} \mid \bar{A})\);(d) \(P(\bar{B} \mid A)\).
If events \(A\) and \(B\) are independent, and \(P(A)=0.45\) and \(P(B)=0.20\), find(a) \(P(A \cap B)\);(b) \(P(A \cup B)\);(c) \(P(\bar{A} \cup \bar{B})\);(d) \(P(B \mid A)\).
The following frequency table shows the classification of 90 students in their sophomore year of college according to their understanding of physics, chemistry, and mathematics.If a student is selected at random, find the probability that the student has (a) an extensive understanding of
Refer to Exercise 3.95. Given that a student, selected at random, is found to have an extensive understanding of physics, what is the probability that the student has(a) an extensive understanding of chemistry;(b) an extensive understanding of both chemistry and mathematics;(c) an extensive
A construction engineer has to inspect 5 construction sites in a 2-day inspection schedule. He may or may not be able to visit these sites in two days. He will not visit any site more than once.(a) Using two coordinates so that \((3,1)\), for example, represents the event that he will visit 3 sites
An explosion in an LNG storage tank in the process of being repaired could have occurred as the result of static electricity, malfunctioning electrical equipment, an open flame in contact with the liner, or purposeful action (industrial sabotage). Interviews with engineers who were analyzing the
Do the assumptions for Bernoulli trials appear to hold? Explain. If the assumptions hold, identify success and the probability of interest.(a) A TV ratings company will use their electronic equipment to check a sample of homes around the city to see whether or not each has a set tuned to the
What conditions for the binomial distribution, if any, fail to hold in the following situations?(a) For each of a company's eight production facilities, record whether or not there was an accident in the past week. The largest facility has three times the number of production workers as the
Use Table 1, or software, to find(a) \(B(8 ; 16,0.40)\);(b) \(b(8 ; 16,0.40)\);(c) \(B(9 ; 12,0.60)\);(d) \(b(9 ; 12,0.60)\);(e) \(\sum_{k=6}^{20} b(k ; 20,0.15)\);(f) \(\sum_{k=6}^{9} b(k ; 9,0.70)\).Data From Table 1 Table 11.1 Time to assess problem when a relay tower breaks down Difficulty
Which conditions for the binomial distribution, if any, fail to hold in the following situations?(a) The number of persons having a cold at a family reunion attended by 30 persons.(b) Among 8 projectors in the department office, 2 do not work properly but are not marked defective. Two are selected
Use Table 1, or software, to find(a) \(B(7 ; 18,0.45)\);(b) \(b(7 ; 18,0.45)\);(c) \(B(8 ; 11,0.95)\);(d) \(b(8 ; 11,0.95)\);(e) \(\sum_{k=4}^{11} b(k ; 11,0.35)\);(f) \(\sum_{k=2}^{4} b(k ; 10,0.30)\).Data From Table 1 Table 11.1 Time to assess problem when a relay tower breaks down Difficulty
Rework the decision problem in Example 7, supposing that only 3 of the 20 hard drives required repairs within the first year.Data From Example 7 EXAMPLE 7 Comparing four detergents using an F test An experiment was designed to study the performance of 4 different detergents for cleaning fuel
Voltage fluctuation is given as the reason for \(80 \%\) of all defaults in nonstabilized equipment in a plant. Use the formula for the binomial distribution to find the probability that voltage fluctuation will be given as the reason for three of the next eight defaults.
If the probability is 0.40 that steam will condense in a thin-walled aluminum tube at 10 atm pressure, use the formula for the binomial distribution to find the probability that, under the stated conditions, steam will condense in 4 of 12 such tubes.
During the assembly of an exhaust valve, sufficient distance must be maintained between the valve tip and the cylinder wall. If \(85 \%\) of valve assemblies have the required distance, use Table 1 or software to find the probabilities that among 20 such valves:(a) at least 15 will have the
The probability that the noise level of a wide-band amplifier will exceed \(2 \mathrm{~dB}\) is 0.05. Use Table 1 or software to find the probabilities that among 12 such amplifiers the noise level of(a) one will exceed \(2 \mathrm{~dB}\);(b) at most two will exceed \(2 \mathrm{~dB}\);(c) two or
A milk processing unit claims that, of the processed milk converted to powdered milk, \(95 \%\) does not spoil. Find the probabilities that among 15 samples of powdered milk(a) all 15 will not spoil;(b) at most 12 will not spoil;(c) at least 9 will not spoil.
A quality-control engineer wants to check whether (in accordance with specifications) \(95 \%\) of the electronic components shipped by his company are in good working condition. To this end, he randomly selects 15 from each large lot ready to be shipped and passes the lot if the selected
A food processor claims that at most \(10 \%\) of her jars of instant coffee contain less coffee than claimed on the label. To test this claim, 16 jars of her instant coffee are randomly selected and the contents are weighed; her claim is accepted if fewer than 3 of the jars contain less coffee
Refer to Exercise 4.2.(a) Determine the cumulative probability distribution \(F(x)\).(b) Graph the probability distribution of \(f(x)\) as a bar chart and below it graph \(F(x)\).Data From Exercise 4.2 4.2 An experiment consists of five draws from a pack of cards. Denoting the outcomes BRRBR,
Four emergency radios are available for rescue workers but one does not work properly. Two randomly selected radios are taken on a rescue mission. Let \(X\) be the number that work properly between the two.(a) Determine the probability distribution \(f(x)\) of \(X\).(b) Determine the cumulative
Prove that \(B(x ; n, p)=1-B(n-x-1 ; n, 1-p)\).
Prove that \(b(x ; n, p)=b(n-x ; n, 1-p)\).
With reference to Exercise 4.5, find an expression for the distribution function \(F(x)\) of the random variable.Data From Exercise 4.5 k 4.5 Given that f(x): = is a probability distribution for 2x a random variable that can take on the values x 0, 1, 2, 3, and 4, find k.
Given that \(f(x)=\frac{k}{2^{x}}\) is a probability distribution for a random variable that can take on the values \(x=\) \(0,1,2,3\), and 4 , find \(k\).
Check whether the following can define probability distributions and explain your answers.(a) \(f(x)=\frac{1}{4}\)for \(x=10,11,12,13\)(b) \(f(x)=\frac{2 x}{5}\)for \(x=0,1,2,3,4,5\)(c) \(f(x)=\frac{x-15}{20}\)for \(x=8,9,10,11,12\)(d) \(f(x)=\frac{1+x^{2}}{61}\)for \(x=0,1,2,3,4,5\)
Determine whether the following can be probability distributions of a random variable which can take on only the values \(1,2,3\), and 4 .(a) \(f(1)=0.19, \quad f(2)=0.27, \quad f(3)=0.27\), and \(f(4)=0.27\)(b) \(f(1)=0.24, \quad f(2)=0.24, \quad f(3)=0.24, \quad\) and \(f(4)=0.24\)(c)
An experiment consists of five draws from a pack of cards. Denoting the outcomes BRRBR, BRRRB,..., for black and red cards and assuming that all 32 outcomes are equally likely, find the probability for the total number of red cards.
Amy commutes to work by two different routes A and B. If she comes home by route A, then she will be home no later than 6 P.M. with probability 0.8, but if she comes home by route B, then she will be home no later than 6 P.M. with probability 0.7. In the past, the proportion of times that Amy chose
During the inspection of a rejected integrated circuit (IC), it was observed that the rejected IC could have an incorrect circuit, it could be bent, or it could have both defects. The probability of having an incorrect circuit is 0.45, the probability of being bent is 0.65, and the probability of
Suppose that a probability of \(\frac{1}{16}\) is assigned to each point of the sample space of part (a) of Exercise 3.1 on page 65 . Find the probability distribution of the total number of units of black and white cement that are adulterated.Data From Exercise 3.1 3.1 A civil engineer suspects
Cumulative Poisson probabilities can be calculated using MINITAB.Output:Poisson with mean \(=1.64\)Find the cumulative Poisson probabilities for \(x=2\) and \(x=3\) when(a) \(\lambda=2.73\);(b) \(\lambda=4.33\). Dialog box: Calc Probability Distribution > Poisson > Choose Cumulative Distribution.
Suppose that the probabilities are, respectively, 0.40,0.40, and 0.20 that in city driving a certain kind of imported car will average less than 22 miles per gallon, anywhere from 22 to 25 miles per gallon, or more than 25 miles per gallon. Find the probability that among 12 such cars tested, 4
As can easily be shown, the probabilities of getting 0 , 1 , or 2 heads with a pair of balanced coins are \(\frac{1}{4}, \frac{1}{2}\), and \(\frac{1}{4}\). What is the probability of getting 2 tails twice, 1 head and 1 tail 3 times, and 2 heads once in 6 tosses of a pair of balanced coins?
Suppose the probabilities are 0.89,0.09, and 0.02 that the finish on a new car will be rated acceptable, easily repairable, or unacceptable. Find the probability that, among 20 cars painted one morning, 17 have acceptable finishes, 2 have repairable finishes, and 1 finish is unacceptable.
Using the same sort of reasoning as in the derivation of the formula for the hypergeometric distribution, we can derive a formula which is analogous to the multi-nomial distribution but applies to sampling without replacement. A set of \(N\) objects contains \(a_{1}\) objects of the first kind,
Simulate tossing a coin.(a) For a balanced coin, generate 100 flips.(b) For a coin with probability of heads 0.8, generate 100 flips.
The probabilities that a quality control team will visit \(0,1,2,3\), or 4 production sites on a single day are 0.15,0.22,0.35,0.21, and 0.07.(a) Simulate the inspection team's visits on 30 days.(b) Repeat the simulation of visits on 30 days a total of 100 times. Estimate the probability that there
Depending on the availability of parts, a company can manufacture 3, 4, 5, or 6 units of a certain item per week with corresponding probabilities of 0.10,0.40, 0.30, and 0.20. The probabilities that there will be a weekly demand for \(0,1,2,3, \ldots\), or 8 units are, respectively,
A manufacturer of smart phones has the following probability distribution for the number of defects per phone:(a) Determine the probability of 2 or more defects.(b) Is a randomly selected phone more likely to have 0 defects or 1 or more defects? x f(x) 0 .89 121 .07 .03 3 .01
Upon reviewing recent use of conference rooms at an engineering consulting firm, an industrial engineer determined the following probability distribution for the number of requests for a conference room per half-day:(a) Currently, the building has two conference rooms. What is the probability that
Refer to Exercise 4.80 and obtain the(a) mean;(b) variance;(c) standard deviation for the number of requests for conference rooms.Data From Exercise 4.80 4.80 Upon reviewing recent use of conference rooms at an engineering consulting firm, an industrial engineer determined the following probability
Determine whether the following can be probability distributions of a random variable that can take on only the values of 0,1 , and 2 :(a) \(f(0)=0.34 \quad f(1)=0.34\) and \(f(2)=0.34\).(b) \(f(0)=0.2 \quad f(1)=0.6\) and \(f(2)=0.2\).(c) \(f(0)=0.7 \quad f(1)=0.4\) and \(f(2)=-0.1\).
Check whether the following can define probability distributions, and explain your answers.(a) \(f(x)=\frac{x}{10}\), for \(x=0,1,2,3,4\).(b) \(f(x)=\frac{1}{3}\), for \(x=-1,0,1\).(c) \(f(x)=\frac{(x-1)^{2}}{4}\), for \(x=0,1,2,3\).
An engineering student correctly answers \(85 \%\) of all questions she attempts. What is the probability that the first incorrect answer was the fourth one?
If the probability is 0.20 that a downtime of an automated production process will exceed 2 minutes, find the probability that 3 of 8 downtimes of the process will exceed 2 minutes using (a) the formula for the binomial distribution; (b) Table 1 or software.
If the probability is 0.90 that a new machine will produce 40 or more chairs, find the probabilities that among 16 such machines(a) 12 will produce 40 or more chairs;(b) at least 10 will produce 40 or more chairs;(c) at most 3 will not produce 40 or more chairs.
In 16 experiments studying the electrical behavior of single cells, 12 use micro-electrodes made of metal and the other 4 use micro-electrodes made from glass tubing. If 2 of the experiments are to be terminated for financial reasons, and they are selected at random, what are the probabilities
As can be easily verified by means of the formula for the binomial distribution, the probabilities of getting 0 , 1,2 , or 3 heads in 3 flips of a coin whose probability of heads is 0.4 are 0.216,0.432,0.288, and 0.064. Find the mean of this probability distribution using(a) the formula that
With reference to Exercise 4.88, find the variance of the probability distribution using(a) the formula that defines \(\sigma^{2}\);(b) the special formula for the variance of a binomial distribution.Data From Exercise 4.88 4.88 As can be easily verified by means of the formula for the binomial
Find the mean and the standard deviation of the distribution of each of the following random variables (having binomial distributions):(a) The number of heads in 440 flips of a balanced coin.(b) The number of 6's in 300 rolls of a balanced die.(c) The number of defectives in a sample of 700 parts
Use the Poisson distribution to approximate the binomial probability \(b(1 ; 100,0.02)\).
With reference to Exercise 4.87, find the mean and the variance of the distribution of the number of microelectrodes made from glass tubing using(a) the probabilities obtained in that exercise;(b) the special formulas for the mean and the variance of a hypergeometric distribution.Data From Exercise
The daily number of orders filled by the parts department of a repair shop is a random variable with \(\mu=142\) and \(\sigma=12\). According to Chebyshev's theorem, with what probability can we assert that on any one day it will fill between 82 and 202 orders?
Records show that the probability is 0.00008 that a truck will have an accident on a certain highway. Use the formula for the Poisson distribution to approximate the probability that at least 5 of 20,000 trucks on that highway will have an accident.
The number of weekly breakdowns of a computer is a random variable having a Poisson distribution with \(\lambda=0.2\). What is the probability that the computer will operate without a breakdown for 3 consecutive weeks?
A manufacturer determines that a big screen HDTV set had probabilities of \(0.8,0.15,0.05\), respectively, of being placed in the categories acceptable, minor defect, or major defect. If 3 HDTVs are inspected,(a) find the probability that 2 are acceptable and 1 is a minor defect;(b) find the
Suppose that the probabilities are \(0.2466,0.3452\), \(0.2417,0.1128,0.0395,0.0111,0.0026\), and 0.0005 that there will be \(0,1,2,3,4,5,6\), or 7 polluting spills in the Great Lakes on any one day. Simulate the numbers of polluting spills in the Great Lakes in 30 days.
A candidate invited for a visit has probability 0.6 of being hired. Let \(X\) be the number of candidates that visit before 2 are hired. Find(a) \(P(X \leq 4)\);(b) \(P(X \geq 5)\).
If among \(n\) objects \(k\) are alike and the others are all distinct, the number of permutations of these \(n\) objects taken all together is \(n ! / k !\).(a) How many permutations are there of the letters of the word class?(b) In how many ways can the television director of Exercise 3.21 fill
Determine the number of ways in which a software professional can choose 4 of 25 laptops to test a newly designed application.
How many ways can a company select 4 candidates to interview from a short list of 12 engineers?
A box of 15 spark plugs contains one that is defective. In how many ways can 4 spark plugs be selected so that(a) the defective one is selected;(b) the defective plug is not selected?
With reference to Exercise 3.25, suppose that three of the spark plugs are defective. In how many ways can 4 spark plugs be selected so that(a) one of the defective plugs is selected;(b) two of the defective plugs are selected;(c) all three defective plugs are selected?Data From Exercise 3.25 A
An engineering student has 6 different ball bearings and 9 different gears. In how many ways can 3 ball bearings and 3 gears be selected for an experiment on friction in machine parts?
(a) Among 880 smart phones sold by a retailer, 72 required repairs under the warranty. Estimate the probability that a new phone, which has just been sold, will require repairs under the warranty. Explain your reasoning.(b) Last year 8,400 students applied for the 6,000 student season tickets
A car rental agency has 19 compact cars and 12 intermediate-size cars. If four of the cars are randomly selected for a safety check, what is the probability of getting two of each kind?
With reference to Exercise 3.34, suppose that the questionnaire filled in by one of the 150 persons is to be double-checked. If it is chosen in such a way that each questionnaire has a probability of \(\frac{1}{150}\) of being selected, find the probabilities that the person(a) lives more than 3
The probabilities that a TV station will receive \(0,1,2,3, \ldots, 8\) or at least 9 complaints after showing a controversial program are, respectively, \(0.01,0.03,0.07,0.15,0.19,0.18,0.14,0.12,0.09\), and 0.02. What are the probabilities that after showing such a program the station will
A rotary plug valve needs to be replaced to repair a machine, and the probabilities that the replacement will be a flange style (low pressure), flange style (high pressure), wafer style, or lug style are \(0.16,0.29,0.26\), and 0.15 . Find the probabilities that the replacement will be(a) a

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