Group Work / Sample Questions for Test 2 on Binomial and Hypergeometric Distributions
These were taken from past exams, pasts quizzes, and your text book.

1. For a binomial random variable, X, where n = 6 and p = 0:45, calculate the following.
[Answer all probabilities with at least 4 decimal places]
(a)
(b)
(c) p(2) = P(X = 2)
(d) P(2 X 4)
(e) P(X > 4)
(f) The probability that there is at least one \failure."
(g) The probability that X falls within 2 standard deviations of the mean.

2. A study reposted that 48% of American adults take at least two sick days per year when they are not really sick. In a random sample 30 of American adults:

(a) What is the probability that exactly 13 of the adults in the sample took at least two sick days when they were really not sick?
(b) What is the probability that at most 13 of the adults in the sample took at lesat two sick days when they were really not sick?
3. Suppose a manager has 10 subordinates, 6 of whom are female, while the other 4 are male. The manager will randomly pick 3 of his subordinates to attend a conference in Hawaii.

(a) What is the probability that all three of the employees chosen to attend the conference are male?
(b) What is the probability that all three of the employees chosen to attend the conference are female?
(c) What is the probability that at least one male and at least one female are chosen?

4. Many primary care doctors feel overworked and burdened by potential lawsuits. In fact, the Physicians' Foundation reported that 60% of all general practice physicians in the US do not recommend medicine as a career (Reuters, Nov. 18, 2008). Let X represent the number of sampled general practice physicians who do not recommend medicine as a career.

(a) Is X best described as a Binomial random variable or a Hypergeometric Random Variable?
(b) In a random sample of 15 primary care physicians, what the the probability that at least one does not recommend medicine as a career?
(c) In a sample of 15 primary care physicians, what is the mean and the standard deviation of the number of physicians who do not recommend medicine as a career?

5. Suppose you are on a committee of 9 people, and an important decision is being decided where you will each vote \yes" or \no," and a decision is reached by a simple majority (i.e. 5 or more votes). Your vote only counts when the remaining 8 members of the committee as split 4-4. Suppose that the remaining 8 members of the committee are equally likely to vote \yes" and \no." Find the probability that your vote counts.

6. Suppose you are involved in an investigation about contaminated cartridges from a particular company. In a shipment of 158 cartridges from this company, 36 were found to be contaminated and the other 122 were \clean." If you randomly select 5 of the 158 cartridges, what is the probability that all 5 will be \clean"?

7. A student it taking a multiple choice quiz with 12 questions. Every question has four choices, only one of which is correct. The student did not study, and randomly guesses the answers to all of the 12 questions.

(a) What is the expected number of questions that this student answers correctly?
(b) What is the expected number of questions that this student answers incorrectly?
(c) What is the standard deviation for the number of questions this student answers correctly?
(d) What is the probability that this student answers exactly 6 questions correctly?
(e) What is the probability that this student answers at most 2 questions correctly?
(f) What is the probability that this student answers more than 1 questions correctly?
(g) The professor states that you need to get at least 11 of the 12 questions correct to get an A on the quiz. What is the probability that this student gets an A on the quiz?

8. (Multiple Choice) About 25% automobiles in Michigan are foreign made and nearly 65% of automobiles in California are foreign made. A sample of 100 automobiles are drawn from michigan and a sample of 200 are drawn from california. Let X stand for the number of foreign made cars in the sample from Michigan and Y for the number of foreign cars in the California sample and Z = X+Y. Then

(a) None of X , Y and Z are Binomial.
(b) X and Y are Binomial and Z is not Binomial.
(c) X, Y and Z are all Binomial,

(d) X and Y are not Binomial but Z is Binomial.

9. About 25% automobiles in Michigan are foreign made and nearly 65% of automobiles in California are foreign made. A sample of 100 automobiles are drawn from michigan and a sample of 200 are drawn from california. Let X stand for the number of foreign made cars in the sample from Michigan and Y for the number of foreign cars in the California sample and Z = X+Y. Then
(a) Find the probability that X equals 28.
(b) Find the probability that Y equals 120.

10. Bob's Used Car Dealership has 20 cars: 6 are foreign made and the others are made in the USA. The owner will randomly select four cars from his lot to display in a commercial.

(a) What is the probability that 3 of the four cars shown in the commercial are made in the USA?
(b) What is the probability that at least 3 of the four cars shown in the commercial are made in the
USA?