91影视

Consider independently rolling two fair dice, one red and the other green. Let A be the event that the red die shows \({\rm{3}}\) dots, B be the event that the green die shows \({\rm{4}}\) dots, and C be the event that the total number of dots showing on the two dice is \({\rm{7}}\). Are these events pairwise independent (i.e., are \({\rm{A}}\) and \({\rm{B}}\) independent events, are \({\rm{A}}\) and \({\rm{C}}\) independent, and are \({\rm{B}}\) and \({\rm{C}}\) independent)? Are the three events mutually independent?

Components arriving at a distributor are checked for defects by two different inspectors (each component is checked by both inspectors). The first inspector detects \({\rm{90\% }}\)of all defectives that are present, and the second inspector does likewise. At least one inspector does not detect a defect on \({\rm{20\% }}\)of all defective components. What is the probability that the following occur?

a. A defective component will be detected only by the first inspector? By exactly one of the two inspectors?

b. All three defective components in a batch escape detection by both inspectors (assuming inspections of different components are independent of one another)?

Consider purchasing a system of audio components consisting of a receiver, a pair of speakers, and a \({\rm{CD}}\) player. Let \({{\rm{A}}_{\rm{1}}}\) be the event that the receiver functions properly throughout the warranty period, \({{\rm{A}}_{\rm{2}}}\) be the event that the speakers function properly throughout the warranty period, and \({{\rm{A}}_{\rm{3}}}\) be the event that the \({\rm{CD}}\) player functions properly throughout the warranty period. Suppose that these events are (mutually) independent with \({\rm{P}}\left( {{{\rm{A}}_{\rm{1}}}} \right){\rm{ = }}{\rm{.95}}\), \({\rm{P}}\left( {{{\rm{A}}_{\rm{2}}}} \right){\rm{ = }}{\rm{.98}}\), and \({\rm{P}}\left( {{{\rm{A}}_{\rm{3}}}} \right){\rm{ = }}{\rm{.80}}\).

a. What is the probability that all three components function properly throughout the warranty period?

b. What is the probability that at least one component needs service during the warranty period?

c. What is the probability that all three components need service during the warranty period?

d. What is the probability that only the receiver needs service during the warranty period?

e. What is the probability that exactly one of the three components needs service during the warranty period?

f. What is the probability that all three components function properly throughout the warranty period but that at least one fails within a month after the warranty expires?

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 is actually present, let p denote the probability that the flaw is detected during any one fixation (this model is discussed in 鈥淗uman Performance in Sampling Inspection,鈥� Human Factors, \({\rm{1979: 99--105)}}{\rm{.}}\)

a. Assuming that an item has a flaw, what is the probability that it is detected by the end of the second fixation (once a flaw has been detected, the sequence of fixations terminates)?

b. Give an expression for the probability that a flaw will be detected by the end of the nth fixation.

c. If when a flaw has not been detected in three fixations, the item is passed, what is the probability that a flawed item will pass inspection?

d. Suppose \({\rm{10\% }}\) of all items contain a flaw (P(randomly chosen item is flawed) . \({\rm{1}}\)). With the assumption of part (c), what is the probability that a randomly chosen item will pass inspection (it will automatically pass if it is not flawed, but could also pass if it is flawed)?

e. Given that an item has passed inspection (no flaws in three fixations), what is the probability that it is actually flawed? Calculate for \({\rm{p = 5}}\).

a. A lumber company has just taken delivery on a shipment of \({\rm{10,000 2 \times 4}}\)boards. Suppose that 20% of these boards (\({\rm{2000}}\)) are actually too green to be used in first-quality construction. Two boards are selected at random, one after the other. Let A 5 {the first board is green} and B 5 {the second board is green}. Compute \({\rm{P(A),P(B)}}\), and \({\rm{P(A}} \cap {\rm{B)}}\) (a tree diagram might help). Are \({\rm{A}}\) and \({\rm{B}}\) independent?

b. With \({\rm{A}}\) and \({\rm{B}}\) independent and \({\rm{P(A) = P(B) = }}{\rm{.2,}}\) what is \({\rm{P(A}} \cap {\rm{B)}}\)? How much difference is there between this answer and \({\rm{P(A}} \cap {\rm{B)}}\)part (a)? For purposes of calculating \({\rm{P(A}} \cap {\rm{B)}}\), can we assume that \({\rm{A}}\)and \({\rm{B}}\) of part (a) are independent to obtain essentially the correct probability?

c. Suppose the shipment consists of ten boards, of which two are green. Does the assumption of independence now yield approximately the correct answer for \({\rm{P(A}} \cap {\rm{B)}}\)? What is the critical difference between the situation here and that of part (a)? When do you think an independence assumption would be valid in obtaining an approximately correct answer to \({\rm{P(A}} \cap {\rm{B)}}\)?

Consider randomly selecting a single individual and having that person test drive \({\rm{3}}\) different vehicles. Define events \({{\rm{A}}_{\rm{1}}}\), \({{\rm{A}}_{\rm{2}}}\), and \({{\rm{A}}_{\rm{3}}}\) by

\({{\rm{A}}_{\rm{1}}}\)=likes vehicle #\({\rm{1}}\)\({{\rm{A}}_{\rm{2}}}\)= likes vehicle #\({\rm{2}}\)\({{\rm{A}}_{\rm{3}}}\)=likes vehicle #\({\rm{3}}\)Suppose that\({\rm{ = }}{\rm{.65,}}\)\({\rm{P(}}{{\rm{A}}_3}{\rm{)}}\)\({\rm{ = }}{\rm{.70,}}\)\({\rm{P(}}{{\rm{A}}_{\rm{1}}} \cup {{\rm{A}}_{\rm{2}}}{\rm{) = }}{\rm{.80,P(}}{{\rm{A}}_{\rm{2}}} \cap {{\rm{A}}_{\rm{3}}}{\rm{) = 40,}}\)and\({\rm{P(}}{{\rm{A}}_{\rm{1}}} \cup {{\rm{A}}_{\rm{2}}} \cup {{\rm{A}}_{\rm{3}}}{\rm{) = }}{\rm{.88}}{\rm{.}}\)

a. What is the probability that the individual likes both vehicle #\({\rm{1}}\)and vehicle #\({\rm{2}}\)?

b. Determine and interpret\({\rm{p}}\)(\({{\rm{A}}_{\rm{2}}}\)|\({{\rm{A}}_{\rm{3}}}\)).

c. Are \({{\rm{A}}_{\rm{2}}}\)and \({{\rm{A}}_{\rm{3}}}\)independent events? Answer in two different ways.

d. If you learn that the individual did not like vehicle #\({\rm{1}}\), what now is the probability that he/she liked at least one of the other two vehicles?

The probability that an individual randomly selected from a particular population has a certain disease is \({\rm{.05}}\). A diagnostic test correctly detects the presence of the disease \({\rm{98\% }}\)of the time and correctly detects the absence of the disease \({\rm{99\% }}\)of the time. If the test is applied twice, the two test results are independent, and both are positive, what is the (posterior) probability that the selected individual has the disease? (Hint: Tree diagram with first-generation branches corresponding to Disease and No Disease, and second- and third-generation branches corresponding to results of the two tests.)

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 \({{\rm{C}}_{\rm{1}}}{\rm{ = }}\){left ear tag is lost} and \({{\rm{C}}_{\rm{2}}}{\rm{ = }}\){right ear tag is lost}. Let 颅 \({\rm{\pi = P(}}{{\rm{C}}_{\rm{1}}}{\rm{) = P(}}{{\rm{C}}_{\rm{2}}}{\rm{)}}\),and assume \({{\rm{C}}_{\rm{1}}}\)and \({{\rm{C}}_{\rm{2}}}\) are independent events. Derive an expression (involving p) for the probability that exactly one tag is lost, given that at most one is lost (鈥淓ar Tag Loss in Red Foxes,鈥� J. Wildlife Mgmt., \({\rm{1976: 164--167)}}{\rm{.}}\) (Hint: Draw a tree diagram in which the two initial branches refer to whether the left ear tag was lost.)

An engineering construction firm is currently working on power plants at three different sites. Let A_idenote the event that the plant at site i is completed by the contract date. Use the operations of union, intersection, and complementation to describe each of the following events in terms of \({A_1}\), \({A_2}\), and \({A_3}\), draw a Venn diagram, and shade the region corresponding to each one.

a. At least one plant is completed by the contract date.

b. All plants are completed by the contract date.

c. Only the plant at site 1 is completed by the contract date.

d. Exactly one plant is completed by the contract date.

e. Either the plant at site 1 or both of the other two plants are completed by the contract date.

Use Venn diagrams to verify the following two relationships for any events Aand B (these are called De Morgan鈥檚 laws):

a.\(\left( {A \cup B} \right)' = A' \cap B'\)

b.\(\left( {A \cap B} \right)' = A' \cup B'\)

Hint:In each part, draw a diagram corresponding to the left side and another corresponding to the right side.)