91Ó°ÊÓ

From a box containing 6 black balls and 4 green balls, 3 balls are drawn in succession, each ball being replaced in the box before the next draw is made. What is the probability that (a) all 3 are the same color? (b) each color is represented?

A certain federal agency employs three consulting firms \((\mathrm{A}, B,\) and \(C)\) with probabilities \(0.40,\) \(0.35,\) and \(0.25,\) respectively. From past experience it is known that the probability of cost overruns for the firms are \(0.05,0.03,\) and \(0.15,\) respectively. Suppose a cost overrun is experienced by the agency. (a) What is the probability that the consulting firm involved is company C? (b) What is the probability that it is company A?

A manufacturer is studying the effects of cooking temperature, cooking time, and type of cooking oil for making potato chips. Three different temperatures, 4 different cooking times, and 3 different oils are to be used. (a) What is the total number of combinations to be studied? (b) How many combinations will be used for each type of oil? (c) Discuss why permutations are not an issue in this exercise.

A certain form of cancer is known to be found in women over 60 with probability 0.07 . A blood test exists for the detection of the disease but the test is not infallible. In fact, it is known that \(10 \%\) of the time the test gives a false negative (i.e., the test incorrectly gives a negative result) and \(5 \%\) of the time the test gives a false positive (i.e., incorrectly gives a positive result). If a woman over 60 is known to have taken the test and received a favorable (i.e., a negative result), what is the probability that she has the disease?

A producer of a certain type of electronic component ships to suppliers in lots of twenty. Suppose that \(60 \%\) of all such lots contain no defective components, \(30 \%\) contain one defective component, and \(10 \%\) contain two defective components. A lot is selected and two components from the lot are randomly selected and tested and neither is defective. (a) What is the probability that zero defective components exist in the lot? (b) What is the probability that one defective exists in the lot? (c) What is the probability that two defectives exist in the lot?

A rare disease exists in which only 1 in 500 are affected. A test for the disease exists but of course it is not infallible. A correct positive result (patient actually has the disease) occurs \(95 \%\) of the time while a false positive result (patient does not have the disease) occurs \(1 \%\) of the time. If a randomly selected individual is tested and the result is positive, what is the probability that the individual has the disease?

A construction company employs 2 sales engineers. Engineer 1 does the work in estimating cost for \(70 \%\) of jobs bid by the company. Engineer 2 does the work for \(30 \%\) of jobs bid by the company. It is known that the error rate for engineer 1 is such that 0.02 is the probability of an error when he does the work, whereas the probability of an error in the work of engineer 2 is 0.04. Suppose a bid arrives and a serious error occurs in estimating cost. Which engineer would you guess did the work? Explain and show all work.

In the field of quality control the science of statistics is often used to determine if a process is "out of control." Suppose the process is, indeed, out of control and \(20 \%\) of items produced are defective. (a) If three items arrive off the process line in succession, what is the probability that all three are defective? (b) If four items arrive in succession, what is the probability that three are defective?

A firm is accustomed to training operators who do certain tasks on a production line. Those operators who attend the training course are known to be able to meet their production quotas \(90 \%\) of the time. New operators who do not take the training course only meet their quotas \(65 \%\). of the time. Fifty percent of new operators attend the course. Given that a new operator meets his production quota, what is the probability that he (or she) attended the program?

A survey of those using a particular statistical software system indicated that \(10 \%\) were dissatisfied. Half of those dissatisfied purchased the system from vender A. It is also known that \(20 \%\) of those surveyed purchased from vendor A. Given that the software package was purchased from vendor \(A\), what is the probability that that particular user is dissatisfied?