Question 1
If
(a)
(b)
(c)
Question 2
In a certain class, 5 students obtained an A, 10 students obtained a B, 17 students obtained a C, and 6 students obtained a D. What is the probability that a randomly chosen student receive a B? If a student receives $10 for an A, $5 for a B, $2 for a C, and $0 for a D, what is the average gain that a student will make from this course?
Question 3
Consider the problem of screening for cervical cancer. The probability that a women has the cancer is 0.0001. The screening test correctly identifies 90% of all the women who do have the disease, but the test is false positive with probability 0.001.
(a)
Find the probability that a woman actually does have cervical cancer given the test says she does.
(b)
List the four possible outcomes in the sample space.
| Has Cancer ( |
Does Not Have Cancer ( |
|
|---|---|---|
| Positive Test ( |
true positive | false positive |
| Negative Test ( |
false negative | true negative |
Question 4
An automobile insurance company classifies each driver as a good risk, a medium risk, or a poor risk. Of those currently insured, 30% are good risks, 50% are medium risks, and 20% are poor risks. In any given year the probability that a driver will have at least one accident is 0.1 for a good risk, 0.3 for a medium risk, and 0.5 for a poor risk.
(a)
What is the probability that the next customer randomly selected will have at least one accident next year?
(b)
If a randomly selected driver insured by this company had an accident this year, what is the probability that this driver was actually a good risk?
Problem 5
A truth serum given to a suspect is known to be 90% reliable when the person is guilty and 99% reliable when the person is innocent. In other words, 10% of the guilty are judged innocent by the serum and 1% of the innocent are judged guilty. If the suspect was selected from a group of suspects of which only 5% have ever committed a crime, and the serum indicates that he is guilty, what is the probability that he is innocent?
Problem 6
70% 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 80% of the aircraft not discovered do not have an emergency locator.
(a)
What percentage of the aircraft have an emergency locator?
(b)
What percentage of the aircraft with emergency locator are discovered after they disappear.
Problem 7
Two methods, A and B, are available for teaching a certain industrial skill. The failure rate is 20% for A and 10% for B. However, B is more expensive and hence is only used 30% of the time (A is used the other 70%). A worker is taught the skill by one of the methods, but fails to learn it correctly. What is the probability that the worker was taught by Method A?
Problem 8
Suppose that the numbers 1 through 10 form the sample space of a random experiment, and assume that each number is equally likely. Define the following events:
(a)
Are
No they are not mutually exclusive as
(b)
Calculate
(b)
Are
Problem 9
A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed three times
(a)
What is the sample space of the random experiment?
(b)
What is the probability of getting exactly two tails?
Problem 10
Items in your inventory are produced at three different plants: 50 percent from plant
You select an item from your inventory and it turns out to be defective. Which plant is the item most likely to have come from? Why does knowing the item is defective decrease the probability that it has come from plant
Knowing that the item is defective decreases the probability it comes from plant
Problem 11
Calculate the reliability of the system described in the following figure. The numbers beside each component represent the probabilities of failure for this component. Note that the components work independently of one another.
Problem 12
A system consists of two subsystems connected in series. Subsystem 1 has two components connected in parallel. Subsystem 2 has only one component. Suppose the three components work independently and each has probability of failure equal to 0.2. What is the probability that the system works?
Problem 13
A proficiency examination for a certain skill was given to 100 employees of a firm. Forty of the employees were male. Sixty of the employees passed the exam, in that they scored above a preset level for satisfactory performance. The breakdown among males and females was as follows:
| Male ( |
Female ( |
|
|---|---|---|
| Pass ( |
24 | 36 |
| Fail | 16 | 23 |
Suppose an employee is randomly selected from the 100 who took the examination.
(a)
Find the probability that the employee passed, given that he was male.
(b)
Find the probability that the employee was male, given that he passed.
(c)
Are the events
Yes they are independent.
(d)
Are the events
Yes they are independent.
Problem 16
A large company hires most of its employees on the basis of two tests. The two tests have scores ranging from one to five. The following table summarizes the performance of 16,839 applicants during the last six years.
| Score | 1 | 2 | 3 | 4 | 5 | Total |
|---|---|---|---|---|---|---|
| 1 | 0.07 | 0.03 | 0.00 | 0.00 | 0.00 | 0.10 |
| 2 | 0.15 | 0.03 | 0.02 | 0.00 | 0.00 | 0.20 |
| 3 | 0.08 | 0.15 | 0.09 | 0.02 | 0.01 | 0.35 |
| 4 | 0.10 | 0.04 | 0.08 | 0.01 | 0.02 | 0.25 |
| 5 | 0.00 | 0.00 | 0.06 | 0.02 | 0.02 | 0.10 |
| Total: | 0.40 | 0.25 | 0.25 | 0.05 | 0.05 | 1.00 |
From this table we learn, for example, that 3% of the applicants got a score of 2 on Test 1 and 2 on Test 2; and that 15% of the applicants got a score of 3 on Test 1 and 2 on Test 2. We also learn that, for example, 20% of the applicants got a score of 2 on Test 1 and that 25% of the applicants got a score of 2 on Test 2.
A group of 1500 new applicants have been selected to take the tests.
(a)
What should the cutting scores be if between 140 and 180 applicants will be short–listed for a job interview? Assume that the company wishes to short–list people with the highest possible performances on the two tests.