Suppose that P(A | B) = 0.2, P(A | B′) = 0.3, and P(B) = 0.8. What is P(A)?
> A cone with height h is inscribed in a larger cone with height H so that its vertex is at the center of the base of the larger cone. Show that the inner cone has maximum volume when h = 1/3H.
> A cone-shaped paper drinking cup is to be made to hold 27 cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper.
> A cone-shaped drinking cup is made from a circular piece of paper of radius R by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup.
> A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?
> A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is (a) a maximum? (b) A minimum?
> A right circular cylinder is inscribed in a cone with height and base radius r. Find the largest possible volume of such a cylinder.
> A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus, the diameter of the semicircle is equal to the width of the rectangle) If the perimeter of the window is 30 ft, find the dimensions of the window so that the greatest possibl
> A cylindrical can without a top is made to contain V cm3 of liquid. Find the dimensions that will minimize the cost of the metal to make the can.
> Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius r.
> Find the area of the largest rectangle that can be inscribed in the ellipse x2/a2 + y2/b2 = 1.
> Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs (with a graphing device) that display the major features of the curve. Use these graphs to estimate the maximum and minimum
> A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible volume of such a cylinder.
> Find the dimensions of the rectangle of largest area that has its base on the -axis and its other two vertices above the x-axis and lying on the parabola y = 8 – x2.
> Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side L if one side of the rectangle lies on the base of the triangle.
> Find, correct to two decimal places, the coordinates of the point on the curve y = tan x that is closest to the point (1, 1).
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> A rectangular storage container with an open top is to have a volume of 10 m3. The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for t
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> A box with a square base and open top must have a volume of 32,000 cm3. Find the dimensions of the box that minimize the amount of material used.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> If 1200 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> The rate (in mg carbon/m3/h) at which photosynthesis takes place for a species of phytoplankton is modeled by the function P = 100I/I2 + I + 4 where I is the light intensity (measured in thousands of foot candles). For what light intensity is P a maximum
> A model used for the yield of an agricultural crop as a function of the nitrogen level N in the soil (measured in appropriate units) is Y = KN/1 + N2 where is a positive constant. What nitrogen level gives the best yield?
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Given that which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible.
> Find two positive numbers whose product is 100 and whose sum is a minimum.
> Find two numbers whose difference is 100 and whose product is a minimum.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. (a)
> Consider the following problem: Find two numbers whose sum is 23 and whose product is a maximum. (a). Make a table of values, like the following one, so that the sum of the numbers in the first two columns is always 23. On the basis of the evidence in yo
> Similar to the hospital schedule in Example 2.9, suppose that an operating room needs to schedule three knee, four hip, and five shoulder surgeries. Assume that all schedules are equally likely. Determine the probability for each of the following: a. All
> A Web ad can be designed from four different colors, three font types, five font sizes, three images, and five text phrases. A specific design is randomly generated by the Web server when you visit the site. If you visit the site five times, what is the
> Suppose that a patient is selected randomly from those described in Exercise 2.1.25. Let A denote the event that the patient is in the group treated with interferon alfa, and let B denote the event that the patient has a complete response. Determine the
> Use the axioms of probability to show the following: a. For any event E, P(E′) = 1 − P(E). b. P(Ø) = 0 c. If A is contained in B, then P(A) ≤ P(B). Answer: (a) Because E and E' are mutually exclusive events and = S 1 = P(S) = P( ) = P(E) + P(E'). The
> Similar to the hospital schedule in Example 2.9, suppose that an operating room needs to schedule three knee, four hip, and five shoulder surgeries. Assume that all schedules are equally likely. Determine the following probabilities: a. All hip surgeries
> AWeb ad can be designed fromfour different colors, three font types, five font sizes, three images, and five text phrases. A specific design is randomly generated by the Web server when you visit the site. Determine the probability that the ad color is r
> Suppose that a patient is selected randomly from those described in Exercise 2.4.11. Let A denote the event that the patient is in group 1, and let B denote the event for which there is no progression. Determine the following probabilities: a. P(A ∩ B)
> Consider the hospital emergency room data in Example 2.6. Let A denote the event that a visit is to hospital 4 and let B denote the event that a visit results in LWBS (at any hospital). Data from Example 2.6: Determine the following probabilities. a. P
> An article in the British Medical Journal [“Comparison of Treatment of Renal Calculi by Operative Surgery, Percutaneous Nephrolithotomy, and Extracorporeal Shock Wave Lithotripsy” (1986, Vol. 82, pp. 879–892)] provided the following discussion of success
> Computer keyboard failures are due to faulty electrical connects (12%) or mechanical defects (88%). Mechanical defects are related to loose keys (27%) or improper assembly (73%). Electrical connect defects are caused by defective wires (35%), improper co
> The following circuit operates if and only if there is a path of functional devices from left to right. Assume devices fail independently and that the probability of failure of each device is as shown. What is the probability that the circuit operates?
> A lot of 100 semiconductor chips contains 20 that are defective. a. Two are selected, at random, without replacement, from the lot. Determine the probability that the second chip selected is defective. b. Three are selected, at random, without replacemen
> The edge roughness of slit paper products increases as knife blades wear. Only 1% of products slit with new blades have rough edges, 3% of products slit with blades of average sharpness exhibit roughness, and 5% of products slit with worn blades exhibit
> Heart failures are due to either natural occurrences (87%) or outside factors (13%). Outside factors are related to induced substances (73%) or foreign objects (27%). Natural occurrences are caused by arterial blockage (56%), disease (27%), and infection
> The probability is 1% that an electrical connector that is kept dry fails during the warranty period. If the connector is ever wet, the probability of a failure during the warranty period is 5%. If 90% of the connectors are kept dry and 10% are wet, what
> Suppose that P(A | B) = 0.4 and P(B) = 0.5. Determine the following: a. P(A ∩ B) b. P(A′ ∩ B)
> Consider the hospital emergency room data in Example 2.6. Let A denote the event that a visit is to hospital 4, and let B denote the event that a visit results in LWBS(at any hospital). Determine the following probabilities. a. P(A ∩ B) b. P(A′) c. P(A
> Suppose that a patient is selected randomly from those described in Exercise 2.4.11. Let A denote the event that the patient is in group 1, and let B denote the event that there is no progression. Determine the following probabilities: a. P(B | A) b. P(
> A computer system uses passwords that contain exactly eight characters, and each character is one of the 26 lowercase letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9). Let Ω denote the set of all possible passwords. Suppose that all passw
> Consider the hospital emergency room data in Example 2.6. Let A denote the event that a visit is to hospital 4, and let B denote the event that a visit results in LWBS (at any hospital). Determine the following probabilities. a. P(A | B) b. P(A′ | B) c.
> Let E1, E2, and E3 denote the samples that conform to a percentage of solids specification, a molecular weight specification, and a color specification, respectively.Atotal of 240 samples are classified by the E1, E2, and E3 specifications, where yes ind
> An article in The Canadian Entomologist (Harcourt et al., 1977, Vol. 109, pp. 1521–1534) reported on the life of the alfalfa weevil from eggs to adulthood. The following table shows the number of larvae that survived at each stage of de
> Suppose A and B are mutually exclusive events. Construct a Venn diagram that contains the three events A, B, and C such that P(A | C) = 1 and P(B | C) = 0.
> A batch of 500 containers for frozen orange juice contains 5 that are defective. Three are selected, at random, without replacement from the batch. a. What is the probability that the second one selected is defective given that the first one was defectiv
> Consider the endothermic reactions in Exercise 2.1.22. Let A denote the event that a reaction’s final temperature is 271 K or less. Let B denote the event that the heat absorbed is above target. Determine the following probabilities. a. P(A | B) b. P(A′
> A maintenance firm has gathered the following information regarding the failure mechanisms for air conditioning systems: The units without evidence of gas leaks or electrical failure showed other types of failure. If this is a representative sample of AC
> The following table summarizes the number of deceased beetles under autolysis (the destruction of a cell after its death by the action of its own enzymes) and putrefaction (decomposition of organic matter, especially protein, by microorganisms, resulting
> Samples of skin experiencing desquamation are analyzed for both moisture and melanin content. The results from 100 skin samples are as follows: Let A denote the event that a sample has low melanin content, and let B denote the event that a sample has hig
> The analysis of results from a leaf transmutation experiment (turning a leaf into a petal) is summarized by type of transformation completed: a. If a leaf completes the color transformation, what is the probability that it will complete the textural tran
> The article “Clinical and Radiographic Outcomes of Four Different Treatment Strategies in Patients with Early Rheumatoid Arthritis” [Arthritis & Rheumatism (2005, Vol. 52, pp. 3381–3390)] considered four treatment groups. The groups consisted of patients
> Consider the three patient groups in Exercise 2.1.25. Let A denote the event that the patient was treated with ribavirin plus interferon alfa, and let B denote the event that the response was complete. Determine the following probabilities: a. P(A ∪ B)
> Transactions to a computer database are either new items or changes to previous items. The addition of an item can be completed in less than 100 milliseconds 90% of the time, but only 20% of changes to a previous item can be completed in less than this t
> A computer system uses passwords that contain exactly eight characters, and each character is one of the 26 lowercase letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9). Assume all passwords are equally likely. Let A and B denote the events
> A Web ad can be designed from four different colors, three font types, five font sizes, three images, and five text phrases. A specific design is randomly generated by the Web server when you visit the site. Let A denote the event that the design color i
> A computer system uses passwords that are six characters, and each character is one of the 26 letters (a–z) or 10 integers (0–9). Uppercase letters are not used. Let A denote the event that a password begins with a vowel (either a, e, i, o, or u), and le
> Strands of copper wire from a manufacturer are analyzed for strength and conductivity. The results from 100 strands are as follows: a. If a strand is randomly selected, what is the probability that its conductivity is high and its strength is high? b. If
> Consider the hospital emergency room data in Example 2.6. Let A denote the event that a visit is to hospital 4, and let B denote the event that a visit results in LWBS (at any hospital). Use the addition rules to calculate the following probabilities. a
> In the article “ACL Reconstruction Using Bone-Patellar Tendon-Bone Press-Fit Fixation: 10-Year Clinical Results” in Knee Surgery, Sports Traumatology, Arthroscopy (2005, Vol. 13, pp. 248–255), the fol
> A manufacturer of front lights for automobiles tests lamps under a high-humidity, high-temperature environment using intensity and useful life as the responses of interest. The following table shows the performance of 130 lamps: a. Find the probability t
> If P(A) = 0.3, P(B) = 0.2, and P(A ∩ B) = 0.1, determine the following probabilities: a. P(A′) b. P(A ∪ B) c. P(A′ ∩ B) d. P(A ∩ B′) e. P[(A ∪ B)′] f. P(A′ ∪ B)
> If A, B, and C are mutually exclusive events with P(A) = 0.2, P(B) = 0.3, and P(C) = 0.4, determine the following probabilities: a. P(A ∪ B ∪ C) b. P(A ∩ B ∩ C) c. P(A ∩ B) d. P[(A ∪ B) ∩ C] e. P(A′ ∩ B′ ∩ C′)
> Samples of emissions from three suppliers are classified for conformance to air-quality specifications. The results from 100 samples are summarized as follows: Let A denote the event that a sample is from supplier 1, and let B denote the event that a sam
> In circuit testing of printed circuit boards, each board either fails or does not fail the test. A board that fails the test is then checked further to determine which one of five defect types is the primary failure mode. Represent the sample space for t
> An article in the Journal of Database Management [“Experimental Study of a Self-Tuning Algorithm for DBMS Buffer Pools” (2005, Vol. 16, pp. 1–20)] provided the workload used in the TPC-C OLTP (Transac
> Magnesium alkyls are used as homogenous catalysts in the production of linear low-density polyethylene (LLDPE), which requires a finer magnesium powder to sustain a reaction. Redox reaction experiments using four different amounts of magnesium powder are
> Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows: Let A denote the event that a disk has high shock resistance, and let B denote the event that a disk has
> Suppose your vehicle is licensed in a state that issues license plates that consist of three digits (between 0 and 9) followed by three letters (between A and Z). If a license number is selected randomly, what is the probability that yours is the one sel
> A message can follow different paths through servers on a network. The sender’s message can go to one of five servers for the first step; each of them can send to five servers at the second step; each of those can send to four servers at the third step;
> In a NiCd battery, a fully charged cell is composed of nickelic hydroxide. Nickel is an element that has multiple oxidation states and that is usually found in the following states: a. What is the probability that a cell has at least one of the positive
> A credit card contains 16 digits between 0 and 9. However, only 100 million numbers are valid. If a number is entered randomly, what is the probability that it is a valid number?
> An injection-molded part is equally likely to be obtained from any one of the eight cavities on a mold. a. What is the sample space? b. What is the probability that a part is from cavity 1 or 2? c. What is the probability that a part is from neither cavi
> A part selected for testing is equally likely to have been produced on any one of six cutting tools. a. What is the sample space? b. What is the probability that the part is from tool 1? c. What is the probability that the part is from tool 3 or tool 5?
> The sample space of a random experiment is {a, b, c, d, e} with probabilities 0.1, 0.1, 0.2, 0.4, and 0.2, respectively. Let A denote the event {a, b, c}, and let B denote the event {c, d, e}. Determine the following: a. P(A) b. P(B) c. P(A′) d. P(A ∪
> The probability that a customer’s order is not shipped on time is 0.05. A particular customer places three orders, and the orders are placed far enough apart in time that they can be considered to be independent events. a. What is the probability that al
> Similar to the hospital schedule in Example 2.9, suppose that an operating room needs to handle three knee, four hip, and five shoulder surgeries. a. How many different sequences are possible? b. How many different sequences have all hip, knee, and shoul
> Plastic parts produced by an injection-molding operation are checked for conformance to specifications. Each tool contains 12 cavities in which parts are produced, and these parts fall into a conveyor when the press opens. An inspector selects 3 parts wi
> A bin of 50 parts contains 5 that are defective. A sample of 10 parts is selected without replacement. How many samples contain at least 4 defective parts?
> In the design of an electromechanical product, 12 components are to be stacked into a cylindrical casing in a manner that minimizes the impact of shocks. One end of the casing is designated as the bottom and the other end is the top. a. If all components
> Consider the design of a communication system. a. How many three-digit phone prefixes that are used to represent a particular geographic area (such as an area code) can be created from the digits 0 through 9? b. As in part (a), how many three-digit phone
> In the layout of a printed circuit board for an electronic product, 12 different locations can accommodate chips. a. If five different types of chips are to be placed on the board, how many different layouts are possible? b. If the five chips that are pl
> In the laboratory analysis of samples from a chemical process, five samples from the process are analyzed daily. In addition, a control sample is analyzed twice each day to check the calibration of the laboratory instruments. a. How many different sequen
> In a sheet metal operation, three notches and four bends are required. If the operations can be done in any order, how many different ways of completing the manufacturing are possible?
> Abatch of 140 semiconductor chips is inspected by choosing a sample of 5 chips without replacement. Assume 10 of the chips do not conform to customer requirements. a. How many different samples are possible? b. How many samples contain exactly one noncon
> A manufacturing process consists of 10 operations that can be completed in any order. How many different production sequences are possible?
> Energy released from cells breaks the molecular bond and converts ATP (adenosine triphosphate) into ADP (adenosine diphosphate). Storage of ATP in muscle cells (even for an athlete) can sustain maximal muscle power only for less than five seconds (a shor
> New designs for a wastewater treatment tank have proposed three possible shapes, four possible sizes, three locations for input valves, and four locations for output valves. How many different product designs are possible?
