1. Bayes' theorem is particularly useful when:

A. Events are mutually exclusive B. You want to "flip" conditional probabilities C. Trying to find probabilities of independent events D. Calculating expected value

2. Given two children, the probability that both are boys given at least one is a boy is:

A. 1/2 B. 1/3 C. 1/4 D. 2/3

3. The number of ways to arrange n distinct people in a line is:

A. \(n\) B. \(n^2\) C. \(n!\) D. \(2^n\)

4. Which of the following statements about injective functions is correct?

A. Two inputs can have the same output B. Each input has a unique output C. The range equals the codomain D. All outputs are possible from inputs

5. The contrapositive of "If p then q" is:

A. If q then p B. If not q then not p C. If not p then not q D. If p then not q

6. According to Bayes' theorem, if you have the prior probability and the likelihood, what does Bayes' theorem allow you to find?

A. The joint probability of two independent events B. The marginal probability of evidence C. The conditional probability of the hypothesis given the evidence D. The expected value of a random variable

7. Which of the following statements about injective functions is true?

A. Every element in the codomain is mapped to by exactly one element in the domain B. No two elements in the domain map to the same element in the codomain C. Every element in the domain maps to the same element in the codomain D. The function must be onto

8. Which of the following best describes a recurrence relation?

A. A formula to calculate probabilities of events B. An equation defining terms of a sequence using previous terms C. A method to count permutations D. A logical inference rule

9. What is the key difference between permutations and combinations?

A. Permutations are unordered, combinations are ordered B. Permutations allow repetition, combinations do not C. Permutations consider order, combinations do not D. Permutations are always larger in number than combinations

10. In set theory, De Morgan's Law states:

A. \((A \cup B)^c = A^c \cap B^c\) B. \((A \cap B)^c = A^c \cup B^c\) C. Both A and B are true D. None of the above

11. If two events E and F are independent, then:

A. \(P(E \cap F) = 0\) B. \(P(E \cap F) = P(E) + P(F)\) C. \(P(E \cap F) = P(E)P(F)\) D. \(P(E \cup F) = 1\)

12. How many distinct permutations are there for the word "MISSISSIPPI"?

A. \(11!\) B. \(\frac{11!}{4!4!2!}\) C. \(\frac{11!}{1!4!4!2!}\) D. \(\frac{11!}{1!4!4!1!}\)

13. What is the key distinction between permutations and combinations?

A. Whether the objects are distinct or not B. Whether order matters or not C. Whether repetition is allowed or not D. Whether the set is finite or infinite

14. When solving a recurrence relation with repeated roots in its characteristic equation, what general form does the solution take?

A. a_n = αr^n B. a_n = (α + βn)r^n C. a_n = α + βn + γn^2 D. a_n = αr_1^n + βr_2^n

15. In a mathematical induction proof, what is the base step?

A. Prove the statement for some arbitrary k B. Prove the statement for k+1 C. Prove the statement for n=0 or n=1 D. Assume the statement is true for n

16. The characteristic equation of the recurrence relation \(a_n = 4a_{n-1} - 4a_{n-2}\) is:

A. \(r^2 - 4r + 4 = 0\) B. \(r^2 + 4r + 4 = 0\) C. \(r^2 - 2r + 4 = 0\) D. \(r^2 - 4r - 4 = 0\)

17. What does the term "bijection" refer to in the context of functions?

A. A function that is either injective or surjective B. A function that is both injective and surjective C. A function that has no inverse D. A function whose domain and codomain are disjoint

18. The formula for combinations of n objects taken r at a time is:

A. \(\frac{n!}{(n-r)!}\) B. \(\frac{n!}{r!(n-r)!}\) C. \(r^n\) D. \(n^r\)

19. What is expected value linearity?

A. E(X + Y) = E(X) + E(Y) regardless of independence B. E(XY) = E(X) * E(Y) if independent C. E(aX) = a^2 * E(X) D. E(X + Y) = E(X) * E(Y)

20. How many distinct arrangements are there of the letters in the word "MISSISSIPPI"?

A. 39,916,800 B. 34,650 C. 11! D. 10!

21. The pigeonhole principle guarantees that in placing n items into m containers, at least one container has:

A. Exactly one item B. At most one item C. Two or more items if \( n > m \) D. No items

22. The formula for permutations of n distinct objects arranged r at a time is:

A. \(\frac{n!}{(n-r)!}\) B. \(\frac{r!}{n!}\) C. \(n^r\) D. \(\frac{n!}{r!}\)

23. Using Bayes' Theorem, what does the formula calculate?

A. Probability of evidence given hypothesis B. Probability of hypothesis given evidence C. Joint probability of event and condition D. Probability of the union of two events

24. How many ways can 8 people be arranged around a circular table?

A. 8! B. 7! C. 6! D. 8

25. Which of the following is true about a tautology in logic?

A. It is never true B. It is sometimes true C. It is always true D. It is sometimes false

26. A function f(x) = x^3 defined on the real numbers is:

A. Neither injective nor surjective B. Injective but not surjective C. Surjective but not injective D. Both injective and surjective

27. Which principle is used to find the number of ways to choose 5 donuts from 8 types with repetition allowed?

A. Pigeonhole principle B. Inclusion-exclusion C. Stars and bars D. Bernoulli's theorem

28. In logic, the expression ¬(p ∧ q) is equivalent to:

A. ¬p ∧ ¬q B. ¬p ∨ ¬q C. p ∨ q D. p ∧ q

29. In combinatorics, what is the meaning of "stars and bars"?

A. A method to calculate permutations with repetition B. A visualization for combinations with repetition C. A method to prove bijections D. A method to solve recurrence relations

30. In set theory, what does the formula |A ∪ B| = |A| + |B| - |A ∩ B| represent?

A. The inclusion-exclusion principle B. The pigeonhole principle C. Distributive law D. De Morgan's Law

31. The sum of the first n odd numbers is:

A. \(n^2\) B. \(2n\) C. \(\frac{n(n+1)}{2}\) D. \(n(n+1)\)

32. Which method is most appropriate to prove a statement that depends on an integer n for all n ≥ 0?

A. Direct proof B. Proof by contrapositive C. Mathematical induction D. Proof by contradiction

33. If a characteristic equation of a recurrence relation is r^2 - 4r + 4 = 0, what are the roots?

A. 2 and 2 (repeated) B. 4 and 0 C. 1 and 4 D. -2 and 2

34. In which situation do you use permutations?

A. Selecting members of a committee B. Arranging books on a shelf C. Selecting ice cream flavors D. Choosing lottery numbers

35. What is the probability that both children are boys given that at least one child in a family of two children is a boy?

A. 1/4 B. 1/3 C. 1/2 D. 2/3

36. What is the total number of ways to arrange 8 people in a circle?

A. 8! B. 7! C. 8^2 D. 8! / 2

37. If a family has two children, what is the sample space given the condition "not both girls"?

A. {BB, BG, GB, GG} B. {BB, BG, GB} C. {BG, GB} D. {BB}

38. What kind of proof uses the assumption that the negation of the statement leads to a contradiction?

A. Direct proof B. Contrapositive C. Proof by contradiction D. Induction

39. In a problem where order does not matter and repetition is allowed, what combinatorial formula should you use?

A. P(n, r) B. C(n + r - 1, r) C. C(n, r) D. n!

40. An event E has \( |E| = 10 \) and sample space \( |S| = 50 \). What is \(P(E)\) assuming equally likely outcomes?

A. 5 B. 0.2 C. 0.5 D. 10

41. In combinations with repetition allowed, the number of ways to select r objects from n types is:

A. \(\binom{n+r-1}{r}\) B. \(\binom{n}{r}\) C. \(n^r\) D. \(\binom{r}{n}\)

42. What principle can be used to solve the problem: "Among 100 people, at least how many were born in the same month?"

A. The pigeonhole principle B. Bayes' theorem C. Recurrence relations D. Permutations with repetition

43. What is the general solution to a recurrence relation with repeated root r?

A. \(a_n = \alpha r^n\) B. \(a_n = \alpha n r^n\) C. \(a_n = (\alpha + \beta n) r^n\) D. \(a_n = \alpha r^{2n} + \beta\)

44. The symmetric difference between sets A and B, denoted A Δ B, is defined as:

A. A ∩ B B. (A ∪ B) - (A ∩ B) C. A ∪ B D. (A ∩ B)^c

45. If \(f\) is bijective, then:

A. \(f\) is not invertible B. \(f\) has an inverse function C. \(f\) is only injective D. \(f\) is only surjective

46. The expected value operator is linear. Which of the following represents this?

A. \(E(X+Y) = E(X) \cdot E(Y)\) B. \(E(X+Y) = E(X) + E(Y)\) C. \(E(X+Y) = E(X) - E(Y)\) D. \(E(X+Y) = max(E(X), E(Y))\)

47. What does the symmetric difference between two sets A and B represent?

A. Elements in both A and B B. Elements only in A C. Elements in exactly one of A or B D. Elements in none of A or B

48. Which of the following represents the number of ways to arrange n distinct objects in order?

A. n! B. C(n, r) C. P(n, r) D. n^r

49. The probability of drawing two aces consecutively without replacement from a standard deck is:

A. \(\frac{4}{52} \times \frac{3}{51}\) B. \(\frac{4}{52} + \frac{4}{52}\) C. \(\frac{1}{13} \times \frac{1}{13}\) D. \(\frac{2}{52} \times \frac{1}{51}\)

50. When would you apply proof by contrapositive?

A. To prove "If p then q" by proving "If not p then not q" B. To prove "If p then q" by proving "If not q then not p" C. To prove "p if and only if q" by proving both directions D. To prove "p or q" by contradiction