In a class of 120 students:
Let \(E\) = takes English, \(M\) = takes Math, and \(S\) = takes Science.
(a) Find \(P(E\cup M\cup S)\).
(b) How many students take none of the three classes?
(c) How many students take exactly one of the three classes?
(d) How many students take exactly two of the three classes?
(e) If a student is chosen at random and is known to take Math, what is the probability that the student also takes English?
(f) Are \(E\) and \(M\) independent? Explain.
(g) Are \(E\), \(M\), and \(S\) mutually independent? Explain.
A department has 8 English majors, 7 Math majors, and 6 Science majors.
A committee of 6 students is chosen.
(a) How many committees are possible?
(b) How many committees have exactly 2 English majors, 2 Math majors, and 2 Science majors?
(c) How many committees have at least one student from each major?
(d) How many committees have more Math majors than English majors?
(e) Suppose the committee must have a president, vice president, and secretary chosen from the 6 committee members. How many ways are there to choose the committee and assign the three roles?
A password has length 8 and uses lowercase letters and digits.
There are 26 lowercase letters and 10 digits.
(a) How many passwords are possible if repetition is allowed?
(b) How many passwords have exactly 3 digits?
(c) How many passwords have at least one digit?
(d) How many passwords have no repeated characters?
(e) How many passwords have exactly 3 digits and no repeated characters?
A student has 12 identical practice problems to distribute among 4 days.
Let \(x_i\) be the number of problems done on day \(i\).
(a) How many schedules satisfy
\[ x_1+x_2+x_3+x_4=12,\quad x_i\ge 0? \](b) How many schedules satisfy
\[ x_1+x_2+x_3+x_4=12,\quad x_i\ge 1? \](c) How many schedules satisfy
\[ x_1+x_2+x_3+x_4=12,\quad x_1\ge 2,\ x_2\ge 1,\ x_3\ge 0,\ x_4\ge 0? \](d) How many schedules have no day with more than 5 problems?
A club has 10 English students, 8 Math students, and 6 Science students.
Four students are chosen uniformly at random without replacement.
(a) What is the probability that exactly 2 are English students?
(b) What is the probability that the group has 1 English student, 1 Math student, and 2 Science students?
(c) What is the probability that the group has at least one student from each subject?
(d) Given that the group has exactly 2 English students, what is the probability that it also has exactly 1 Math student?
A box contains 4 red balls, 3 blue balls, and 5 green balls.
Two balls are drawn without replacement.
Let \(X\) be the number of red balls drawn.
(a) Find the PMF of \(X\).
(b) Compute \(E[X]\).
(c) Compute \(\mathrm{Var}(X)\).
(d) Is \(X\) binomial? Explain why or why not.
A quiz has 6 multiple-choice questions. Each question has 4 answer choices.
A student knows the answer to each question independently with probability \(0.7\). If they know the answer, they get it correct. If they do not know it, they guess randomly.
Let \(X\) be the number of correct answers.
(a) What is the probability that the student gets a particular question correct?
(b) What distribution does \(X\) have?
(c) Find \(P(X=5)\).
(d) Find \(E[X]\) and \(\mathrm{Var}(X)\).
Let \(X\sim \mathrm{Binomial}(3,0.5)\) and \(Y\sim \mathrm{Binomial}(3,0.5)\).
Assume \(X\) and \(Y\) are independent.
(a) Write the PMFs of \(X\) and \(Y\).
(b) Compute
\[ P(X=Y) \]Let \(X\sim \mathrm{Geometric}(p)\), where \(X\) is the number of trials until the first success.
Assume the first \(m\) trials have already failed.
(a) Compute
\[ P(X>m+n\mid X>m) \](b) Show that
\[ P(X>m+n\mid X>m)=P(X>n) \]This is called the memoryless property.
(c) Compute the expected total number of trials until first success, given that the first \(m\) trials failed:
\[ E[X\mid X>m] \](d) Prove
\[ E[X\mid X>m]=m+\frac{1}{p} \]