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Questions and Answers of
Statistics
As of April 2006, roughly 50 million .com web domain names were registered (e.g., yahoo.com). a. How many domain names consisting of just two letters in sequence can be formed? How many domain names
Three components are connected to form a system as shown in the accompanying diagram. Because the components in the 2-3 subsystem are connected in parallel, that subsystem will function if at least
A friend of mine is giving a dinner party. His current wine supply includes 8 bottles of zinfandel, 10 of merlot, and 12 of cabernet (he only drinks red wine), all from different wineries.a. If he
The composer Beethoven wrote 9 symphonies, 5 piano concertos (music for piano and orchestra), and 32 piano sonatas (music for solo piano). a. How many ways are there to play first a Beethoven
An electronics store is offering a special price on a complete set of components (receiver, compact disc player, speakers, turntable). A purchaser is offered a choice of manufacturer for each
Again consider a Little League team that has 15 players on its roster. a. How many ways are there to select 9 players for the starting lineup? b. How many ways are there to select 9 players for the
Computer keyboard failures can be attributed to electrical defects or mechanical defects. A repair facility currently has 25 failed keyboards, 6 of which have electrical defects and 19 of which have
A production facility employs 10 workers on the day shift, 8 workers on the swing shift, and 6 workers on the graveyard shift. A quality control consultant is to select 5 of these workers for
An academic department with five faculty members narrowed its choice for department head to either candidate A or candidate B. Each member then voted on a slip of paper for one of the candidates.
An experimenter is studying the effects of temperature, pressure, and type of catalyst on yield from a certain chemical reaction. Three different temperatures, four different pressures, and five
A sonnet is a 14-line poem in which certain rhyming patterns are followed. The writer Raymond Queneau published a book containing just 10 sonnets, each on a different page. However, these were
A box in a supply room contains 15 compact fluorescent lightbulbs, of which 5 are rated 13-watt, 6 are rated 18-watt, and 4 are rated 23-watt. Suppose that three of these bulbs are randomly
Each of a sample of four home mortgages is classified as fixed rate (F) or variable rate (V). a. What are the 16 outcomes in S? b. Which outcomes are in the event that exactly three of the selected
Three molecules of type A, three of type B, three of type C, and three of type D are to be linked together to form a chain molecule. One such chain molecule is ABCDABCDABCD, and another is
An ATM personal identification number (PIN) consists of four digits, each a 0, 1, 2,... 8, or 9, in succession. a. How many different possible PINs are there if there are no restrictions on the
A starting lineup in basketball consists of two guards, two forwards, and a center. a. A certain college team has on its roster three centers, four guards, four forwards, and one individual (X) who
In five-card poker, a straight consists of five cards with adjacent denominations (e.g., 9 of clubs, 10 of hearts, jack of hearts, queen of spades, and king of clubs). Assuming that aces can be high
The population of a particular country consists of three ethnic groups. Each individual belongs to one of the four major blood groups. The accompanying joint probability table gives the proportions
Return to the credit card scenario of Exercise 12 (Section 2.2), and let C be the event that the selected student has an American Express card. In addition to P(A) = .6, P(B) = .4, and P(A ⋂ B) =
Reconsider the system defect situation described in Exercise 26 (Section 2.2). In Exercise 26 A certain system can experience three different types of defects. Let Ai (i = 1,2,3) denote the event
The accompanying table gives information on the type of coffee selected by someone purchasing a single cup at a particular airport kiosk. ______________Small ________Medium ________Large Regular
A family consisting of three persons-A, B, and C-goes to a medical clinic that always has a doctor at each of stations 1, 2, and 3. During a certain week, each member of the family visits the clinic
A department store sells sport shirts in three sizes (small, medium, and large), three patterns (plaid, print, and stripe), and two sleeve lengths (long and short). The accompanying tables give the
According to a July 31, 2013, posting on cnn.com subsequent to the death of a child who bit into a peanut, a 2010 study in the journal Pediatrics found that 8% of children younger than 18 in the
A system consists of two identical pumps, #1 and #2. If one pump fails, the system will still operate. However, because of the added strain, the remaining pump is now more likely to fail than was
A certain shop repairs both audio and video components. Let A denote the event that the next component brought in for repair is an audio component, and let B be the event that the next component is a
In Exercise 13, Ai = {awarded project i}, for i = 1, 2, 3. Use the probabilities given there to compute the following probabilities, and explain in words the meaning of each one. a. P(A2 | A1) b.
Deer ticks can be carriers of either Lyme disease or human granulocytic ehrlichiosis (HGE). Based on a recent study, suppose that 16% of all ticks in a certain location carry Lyme disease, 10% carry
For any events A and B with P(B) > 0, show that P(A | B) + P(Aʹ | B) = 1.
If P(B|A) > P(B), show that P(Bʹ | A) < P(Bʹ). [Add P(Bʹ | A) to both sides of the given inequality and then use the result of Exercise 56.]
Show that for any three events A, B, and C with P(C) > 0, P(A ⋃ B|C) = P(A|C) + P(B|C) - P(A ⋂ B|C).
At a certain gas station, 40% of the customers use regular gas (A1), 35% use plus gas (A2), and 25% use premium (A3). Of those customers using regular gas, only 30% fill their tanks (event B). Of
A college library has five copies of a certain text on reserve. Two copies (1 and 2) are first printings, and the other three (3, 4, and 5) are second printings. A student examines these books in
Seventy percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, 60% have an emergency locator, whereas 90%
Components of a certain type are shipped to a supplier in batches of ten. Suppose that 50% of all such batches contain no defective components, 30% contain one defective component, and 20% contain
Blue Cab operates 15% of the taxis in a certain city, and Green Cab operates the other 85%. After a nighttime hit-and-run accident involving a taxi, an eyewitness said the vehicle was blue. Suppose,
For customers purchasing a refrigerator at a certain appliance store, let A be the event that the refrigerator was manufactured in the U.S., B be the event that the refrigerator had an icemaker, and
The Reviews editor for a certain scientific journal decides whether the review for any particular book should be short (1-2 pages), medium (3-4 pages), or long (5-6 pages). Data on recent reviews
A large operator of timeshare complexes requires anyone interested in making a purchase to first visit the site of interest. Historical data indicates that 20% of all potential purchasers select a
Consider the following information about travelers on vacation (based partly on a recent Travelocity poll): 40% check work email, 30% use a cell phone to stay connected to work, 25% bring a laptop
There has been a great deal of controversy over the last several years regarding what types of surveillance are appropriate to prevent terrorism. Suppose a particular surveillance system has a 99%
A friend who lives in Los Angeles makes frequent consulting trips to Washington, D.C.; 50% of the time she travels on airline #1, 30% of the time on airline #2, and the remaining 20% of the time on
In Exercise 59, consider the following additional information on credit card usage:70% of all regular fill-up customers use a credit card.50% of all regular non-fill-up customers use a credit
An academic department has just completed voting by secret ballot for a department head. The ballot box contains four slips with votes for candidate A and three slips with votes for candidate B.
Reconsider the credit card scenario of Exercise 47 (Section 2.4), and show that A and B are dependent first by using the definition of independence and then by verifying that the multiplication
An oil exploration company currently has two active projects, one in Asia and the other in Europe. Let A be the event that the Asian project is successful and B be the event that the European project
In Exercise 13, is any Ai independent of any other Aj? Answer using the multiplication property for independent events. In Exercise 13 A computer consulting firm presently has bids out on three
If A and B are independent events, show that A9 and B are also independent. [First establish a relationship between P(Aʹ ⋂ B), P(B), and P(A ⋂ B).]
The proportions of blood phenotypes in the U.S. population are as follows: A _______B _______AB _______O .40 ....... .11 ........... .04 .......... .45 Assuming that the phenotypes of two randomly
One of the assumptions underlying the theory of control charting (see Chapter 16) is that successive plotted points are independent of one another. Each plotted point can signal either that a
In October, 1994, a flaw in a certain Pentium chip installed in computers was discovered that could result in a wrong answer when performing a division. The manufacturer initially claimed that the
An aircraft seam requires 25 rivets. The seam will have to be reworked if any of these rivets is defective. Suppose rivets are defective independently of one another, each with the same
A boiler has five identical relief valves. The probability that any particular valve will open on demand is .96. Assuming independent operation of the valves, calculate P(at least one valve opens)
Two pumps connected in parallel fail independently of one another on any given day. The probability that only the older pump will fail is .10, and the probability that only the newer pump will fail
An engineering construction firm is currently working on power plants at three different sites. Let Ai denote the event that the plant at site i is completed by the contract date. Use the operations
Consider the system of components connected as in the accompanying picture. Components 1 and 2 are connected in parallel, so that subsystem works iff either 1 or 2 works; since 3 and 4 are connected
Refer back to the series-parallel system configuration introduced in Example 2.36, and suppose that there are only two cells rather than three in each parallel subsystem [in Figure 2.14(a), eliminate
Consider independently rolling two fair dice, one red and the other green. Let A be the event that the red die shows 3 dots, B be the event that the green die shows 4 dots, and C be the event that
Components arriving at a distributor are checked for defects by two different inspectors (each component is checked by both inspectors). The first inspector detects 90% of all defectives that are
Consider purchasing a system of audio components consisting of a receiver, a pair of speakers, and a CD player. Let A1 be the event that the receiver functions properly throughout the warranty
A quality control inspector is examining newly produced items for faults. The inspector searches an item for faults in a series of independent fixations, each of a fixed duration. Given that a flaw
a. A lumber company has just taken delivery on a shipment of 10,000 2 × 4 boards. Suppose that 20% of these boards (2000) are actually too green to be used in first-quality construction. Two boards
Consider randomly selecting a single individual and having that person test drive 3 different vehicles. Define events A1, A2, and A3 by A1 = likes vehicle #1 A2 = likes vehicle #2 A3 = likes vehicle
The probability that an individual randomly selected from a particular population has a certain disease is .05. A diagnostic test correctly detects the presence of the disease 98% of the time and
Suppose identical tags are placed on both the left ear and the right ear of a fox. The fox is then let loose for a period of time. Consider the two events C1 = {left ear tag is lost} and C2 = {right
Use Venn diagrams to verify the following two relationships for any events A and B (these are called De Morgan's laws): a. (A ⋃ B)ʹ = Aʹ ⋂ Bʹ b. (A ⋂ B)ʹ = Aʹ ⋃ Bʹ
A certain legislative committee consists of 10 senators. A subcommittee of 3 senators is to be randomly selected. a. How many different such subcommittees are there? b. If the senators are ranked 1,
A factory uses three production lines to manufacture cans of a certain type. The accompanying table gives percentages of nonconforming cans, categorized by type of nonconformance, for each of the
An employee of the records office at a certain university currently has ten forms on his desk awaiting processing. Six of these are withdrawal petitions and the other four are course substitution
One satellite is scheduled to be launched from Cape Canaveral in Florida, and another launching is scheduled for Vandenberg Air Force Base in California. Let A denote the event that the Vandenberg
A transmitter is sending a message by using a binary code, namely, a sequence of 0's and 1's. Each transmitted bit (0 or 1) must pass through three relays to reach the receiver. At each relay, the
Individual A has a circle of five close friends (B, C, D, E, and F). A has heard a certain rumor from outside the circle and has invited the five friends to a party to circulate the rumor. To begin,
According to the article "Optimization of Distribution Parameters for Estimating Probability of Crack Detection" (J. of Aircraft, 2009: 2090-2097), the following "Palmberg" equation is commonly used
A chemical engineer is interested in determining whether a certain trace impurity is present in a product. An experiment has a probability of .80 of detecting the impurity if it is present. The
Five friends-Allison, Beth, Carol, Diane, and Evelyn-have identical calculators and are studying for a statistics exam. They set their calculators down in a pile before taking a study break and then
Fasteners used in aircraft manufacturing are slightly crimped so that they lock enough to avoid loosening during vibration. Suppose that 95% of all fasteners pass an initial inspection. Of the 5%
The probability that a randomly selected box of a certain type of cereal has a particular prize is .2. Suppose you purchase box after box until you have obtained two of these prizes. a. What is the
A family decides to have children until it has three children of the same gender. Assuming P(B) = P(G) = .5, what is the pmf of X = the number of children in the family?
Three brothers and their wives decide to have children until each family has two female children. What is the pmf of X = the total number of male children born to the brothers? What is E(X), and how
According to the article "Characterizing the Severity and Risk of Drought in the Poudre River, Colorado" (J. of Water Res. Planning and Mgmnt., 2005: 383-393), the drought length Y is the number of
The article "Expectation Analysis of the Probability of Failure for Water Supply Pipes" (J. of Pipeline Systems Engr. and Practice, May 2012: 36-46) proposed using the Poisson distribution to model
Each time a component is tested, the trial is a success (S) or failure (F). Suppose the component is tested repeatedly until a success occurs on three consecutive trials. Let Y denote the number of
Let X be the number of material anomalies occurring in a particular region of an aircraft gas-turbine disk. The article "Methodology for Probabilistic Life Prediction of Multiple-Anomaly Materials"
Suppose that the number of drivers who travel between a particular origin and destination during a designated time period has a Poisson distribution with parameter µ = 20 (suggested in the article
Consider writing onto a computer disk and then sending it through a certifier that counts the number of missing pulses. Suppose this number X has a Poisson distribution with parameter µ = .2.
An article in the Los Angeles Times (Dec. 3, 1993) reports that 1 in 200 people carry the defective gene that causes inherited colon cancer. In a sample of 1000 individuals, what is the approximate
The Centers for Disease Control and Prevention reported in 2012 that 1 in 88 American children had been diagnosed with an autism spectrum disorder (ASD).1. If a random sample of 200 American children
The Centers for Disease Control and Prevention reported in 2012 that 1 in 88 American children had been diagnosed with an autism spectrum disorder (ASD). a. If a random sample of 200 American
Suppose small aircraft arrive at a certain airport according to a Poisson process with rate α = 8 per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with
Organisms are present in ballast water discharged from a ship according to a Poisson process with a concentration of 10 organisms/m3 [the article "Counting at Low Concentrations: The Statistical
The number of requests for assistance received by a towing service is a Poisson process with rate α = 4 per hour. a. Compute the probability that exactly ten requests are received during a
In proof testing of circuit boards, the probability that any particular diode will fail is .01. Suppose a circuit board contains 200 diodes. a. How many diodes would you expect to fail, and what is
The article "Reliability-Based Service-Life Assessment of Aging Concrete Structures" (J. Structural Engr., 1993: 1600-1621) suggests that a Poisson process can be used to represent the occurrence of
An individual named Claudius is located at the point 0 in the accompanying diagram.Using an appropriate randomization device (such as a tetrahedral die, one having four sides), Claudius first moves
Let X have a Poisson distribution with parameter m. Show that E(X) = µ directly from the definition of expected value.
Suppose that trees are distributed in a forest according to a two-dimensional Poisson process with parameter a, the expected number of trees per acre, equal to 80. a. What is the probability that in
Automobiles arrive at a vehicle equipment inspection station according to a Poisson process with rate α = 10 per hour. Suppose that with probability .5 an arriving vehicle will have no equipment
a. In a Poisson process, what has to happen in both the time interval (0, t) and the interval (t, t + Î t) so that no events occur in the entire interval (0, t + Ît)? Use this
Consider a deck consisting of seven cards, marked 1, 2,..., 7. Three of these cards are selected at random. Define an rv W by W = the sum of the resulting numbers, and compute the pmf of W. Then
After shuffling a deck of 52 cards, a dealer deals out 5. Let X = the number of suits represented in the five-card hand.a. Show that the pmf of X is[p(1) = 4P(all are spades), p(2) = 6P(only spades
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