Stat 3201: Practice Midterm 1
0. (Required) Write out the following statement and sign and date. “I have read the exam instructions and agree to adhere to them.
1. [10 points total] Researchers randomly sampled local area students and asked whether they typically packed lunch to bring to school, rather than buying lunch at the school cafeteria. In their sample, 40% of students were in elementary school (ages 5-10, approximately); 20% were in middle school (ages 11-13, approxi- mately); and the rest were in high school (ages 14-18, approximately). 50% of elementary school students typically packed lunch, while 20% of middle school students and 80% of high school students did.
Suppose we randomly select a student from the group that was surveyed; answer the following questions. Be sure to define any symbols you introduce, and use set notation and probability notation where appropriate in your answers.
(a) [5 points] What is the probability that they pack their lunch?
(b) [3 points] What is the probability that they are a middle school student given that they pack their lunch?
(c) [2 points] What is the probability that they are a middle school student given that they do not pack their lunch?
2. [5 points total] Every week, a student group runs a raffle in which 100 tickets are sold and one winner is selected at random. The winner receives a pet rock. I plan to enter the raffle every week until I win. (Once I win, I will stop playing.) Let Y be the (first) week in which I win the pet rock; Y is a random variable.
(a) [3 points] What is the probability mass function of Y? Please justify your answer using counting rules and/or properties of probabilities (i.e., do not just invoke a “named” distribution) and state any assumptions you make.
(b) [2 points] What is the probability that I win the pet rock within the first 3 weeks?
3. [10 points total] Answer the following questions, including a (brief) justification of your answer.
(a) [3 points] A group of five undergraduate and six graduate students are available to serve as represen- tatives to the faculty council. If two students are selected at random, what is the probability that both students are of the same rank (i.e., both undergraduates or both graduates)?
(b) [2 points] Two shoes are selected at random from a tub full of many pairs of shoes (all mixed together). Let:
A = {both selected shoes are right shoes}
B = {both selected shoes are left shoes}
Are A and B independent events? Explain.
(c) [3 points] Let X be a random variable with probability mass function given below:
|
x
|
0
|
1
|
2
|
3
|
|
p(x)
|
0.3
|
c
|
0.4
|
0.1
|
where c is unknown. Find the value c and calculate E[X].
(d) [2 points] Let A and B be events in a sample space S, with P(A) = 0.8 and P(B) = 0.3. Can A and B bemutually exclusive events? Explain.
4. [5 points total] Consider a sample of 200 students, and consider the following sets:
A = {students who are math majors}
B = {students who play ultimate frisbee}
C = {students who have read the Harry Potter books}
Assume |A| = 40, |B| = 80, |C| = 145, |A ∩ B| = 30, |B ∩ C| = 50, |A ∩ C| = 25, and |A ∩ B ∩ C| = 20.
(a) [2 points] Find |(A ∪ B ∪ C)C|.
(b) [3 points] Suppose a student is selected at random from the sample. What is P(AC | BC)?