Questions and Answers of Introduction To Probability Statistics

Show that for 48 million draws from a fair deck of cards, the probability is at least 0.9375 that the proportion of spades drawn will fall between 0.24975 and 0.25025 .
The time taken by students to fill out a loan request form has standard deviation 1.2 hours. What does Chebyshev’s theorem tell us about the probability that a students’ time will be within 4
Prove that (a) = E(X2)-; (b) 33 +2.
Prove that for the Poisson distribution\[\frac{f(x+1 ; \lambda)}{f(x ; \lambda)}=\frac{\lambda}{x+1}\]for \(x=0,1,2, \ldots\)
In a factory, \(8 \%\) of all machines break down at least once a year. Use the Poisson approximation to the binomial distribution to determine the probabilities that among 25 machines (randomly
During inspection of the continuous process of making large rolls of floor coverings, 0.5 imperfections are spotted per minute on average. Use the Poison distribution to find the probabilities(a) one
The number of gamma rays emitted per second by a certain radioactive substance is a random variable having the Poisson distribution with \(\lambda=5.8\). If a recording instrument becomes inoperative
A consulting engineer receives, on average, 0.7 requests per week. If the number of requests follows a Poisson process, find the probability that(a) in a given week, there will be at least 1
A conveyor belt conveys finished products to the warehouse at an average of 2 per minute. Find the probabilities that(a) at most 3 will be conveyed in a given minute;(b) at least 2 will be conveyed
Environmental engineers, concerned about the effects of releasing warm water from a power plants' cooling system into a Great Lake, decided to sample many organisms both inside and outside of a warm
An automated weight monitor can detect underfilled cans of beverages with probability 0.98. What is the probability it fails to detect an underfilled can for the first time when it encounters the
A company manufactures hydraulic gears, and records show that the probability is 0.04 that one of its new gears will fail its inspection test. What is the probability that the fifth gear in a day
Referring to Exercise 4.63, find the probability that the 15th gear in a day is the fourth to fail the test.Data From Exercise 4.63 4.63 A company manufactures hydraulic gears, and records show that
During an assembly process, parts arrive just as they are needed. However, at one station, the probability is 0.01 that a defective part will arrive in a one-hour period. Find the probability that(a)
The arrival of trucks at a receiving dock is a Poisson process with a mean arrival rate of 2 per hour.(a) Find the probability that exactly 5 trucks arrive in a two-hour period.(b) Find the
The number of flaws in a fiber optic cable follows a Poisson process with an average of 0.6 per 100 feet.(a) Find the probability of exactly 2 flaws in a 200 foot cable.(b) Find the probability of
Differentiating with respect to \(p\) on both sides of the equation\[\sum_{x=1}^{\infty} p(1-p)^{x-1}=1\]show that the geometric distribution\[f(x)=p(1-p)^{x-1} \quad \text { for } x=1,2,3,
Poisson probabilities can be calculated using MINITAB.Output:Poisson with mean \(=1.64\)Find the Poisson probabilities for \(x=2\) and \(x=3\) when(a) \(\lambda=2.73\);(b) \(\lambda=4.33\). Dialog
A civil engineer suspects that the cement being supplied for constructing a dam is adulterated. There are three units of white cement and three units of black cement. He will check all six for
With reference to Exercise 3.1, which of the three pairs of events, \(A\) and \(B, B\) and \(C\), and \(B\) and \(C\), are mutually exclusive?Data From Exercise 3.1 3.1 A civil engineer suspects that
With reference to Exercise 3.1, list the outcomes comprising each of the following events, and also express the events in words.(a) \(A \cup B\)(b) \(B \cap C\)(c) \(\bar{B}\)Data From Exercise 3.1
With reference to the sample space of Figure 3.1, express each of the following events in words.(a) \(F=\{(1,0),(1,1)\}\)(b) \(G=\{(0,2),(1,1),(2,0)\}\)(c) \(F \cap G\)Data From Figure 3.1 Figure
To construct sample spaces for experiments in which we deal with non-numerical data, we often code the various alternatives by assigning them numbers. For instance, if an engineer is asked to rate
With reference to Exercise 3.5, which of the pairs of events, \(P\) and \(Q, Q\) and \(R, R\) and \(S\), and \(P\) and \(S\), are mutually exclusive?Data From Exercise 3.5 3.5 To construct sample
Four supervisors and 3 engineers are responsible for work at a construction site, and at least 2 supervisors and one engineer have to be present at all times.(a) Using two coordinates so that
For each of the following experiments, decide whether it would be appropriate to use a sample space which is finite, countably infinite, or continuous.(a) A Geiger counter, located adjacent to a
With reference to Exercise 3.9, what events are represented by(a) regions 1 and 3 together;(b) regions 3 and 4 together;(c) regions 1,2, and 3 together?Data From Exercise 3.9 3.9 In Figure 3.6, C
With reference to Figure 3.4, what events are represented by(a) region 5;(b) regions 4 and 6 together;(c) regions 7 and 8 together;(d) regions 1, 2, 3 and 5 together?Data From Figure 3.4 Figure 3.4
With reference to Figure 3.4, what regions or combinations of regions represent the events that a motor will have(a) none of the major defects;(b) a shaft that is large and windings improper;(c) a
Use Venn diagrams to verify that(a) \(\overline{A \cup B}=\bar{A} \cap \bar{B}\)(b) \(B \cap(A \cup B)=B\)(c) \((A \cup B) \cap(\bar{A} \cup B)=B\)(d) \(A \cap B=(A \cup B) \cap(\bar{A} \cup B)
A building inspector has to check the wiring in a new apartment building either on Monday, Tuesday, Wednesday, or Thursday, and at 8 A.M., 1 P.M., or 2 P.M. A tree diagram which shows the various
If the five finalists in an international volleyball tournament are Spain, the United States, Uruguay, Portugal, and Japan, draw a tree diagram that shows the various possible first- and second-place
If a number cannot be immediately repeated, how many different three number combinations are possible for a combination lock with numbers \(0,1, \ldots, 29\).
Students are offered three cooperative training programs at local companies and four training programs outside the state. Count the number of possible training opportunities if an opportunity
You are required to choose a four digit personal identification number (PIN) for a new debit card. Each digit is selected from \(0,1, \ldots, 9\). How many choices do you have.
An Engineers Association consists of 5 civil engineers and 5 mechanical engineers.(a) In how many ways can a committee of 3 civil engineers and 2 mechanical engineers be appointed?(b) If 2 civil
If there are 9 cars in a race, in how many different ways can they place first, second, and third?
In how many ordered ways can a television director schedule 6 different commercials during the 6 time slots allocated to commercials during the telecast of the first period of a hockey game?
Last year; the maximum daily temperature in a plants' server room exceeded \(68^{\circ} \mathrm{F}\) in 12 days. Estimate the probability that the maximum temperature will exceed \(68^{\circ}
Among 150 persons interviewed as part of an urban mass transportation study, some live more than 3 miles from the center of the city \((A)\), some now regularly drive their own car to work \((B)\),
Refer to parts(d) and(c) of Exercise 3.13 to show that(a) \(P(A \cap B) \leq P(A)\);(b) \(P(A \cup B) \geq P(A)\).Data From Exercise 3.13 3.13 Use Venn diagrams to verify that (a) AUB AnB (b) BN
Explain why there must be a mistake in each of the following statements:(a) The probability that a student will get an \(A\) in a geology course is 0.3, and the probability that he or she will get
Use the definition of Exercise 3.51 to show that if the odds for the occurrence of event A are a to b, where a and b are positive integers, thenData From Exercise 3.51 P = a a+b
Subjective probabilities may or may not satisfy the third axiom of probability. When they do, we say that they are consistent; when they do not, they ought not to be taken too seriously.(a) The
With reference to Figure 3.8, find P(I |D) and P(I |D ), assuming that originally each of the 500 machine parts has the same chance of being chosen for inspection.Data From Figure 3.8 I D 20 10 5
(a) Would you expect the probability that a randomly selected car will need major repairs in the next year to be smaller, remain the same, or increase if you are told it already has high mileage?
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
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
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)
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
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
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
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,
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
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
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
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
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
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
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
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
(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
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)
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
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 \(B\)
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
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
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
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
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)
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
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
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
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 ;
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
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 ;
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
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
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
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
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
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
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
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
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
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
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

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