Counting, induction and discrete probability: basics of counting, pigeonhole principle, permutations and combinations, inclusion–exclusion principle, mathematical induction, probability, Bayes' theorem. - One Line Questions

1. Using the Inclusion-Exclusion Principle for two sets A and B, what is the formula for |A U B|? — |A| + |B| - |A ∩ B|
2. What is the probability of an event that is certain to occur? — 1
3. What is the probability of an event that is impossible to occur? — 0
4. What is the value of C(n, n)? — 1
5. What is the value of P(n, 1)? — n
6. A survey of 100 people found that 70 like coffee and 80 like tea. If 60 people like both, how many like neither? — 10
7. If P(A) = 0.6, P(B) = 0.5, P(A U B) = 0.8, what is P(A ∩ B)? — 0.3
8. If P(A) = 0.6 and P(B) = 0.7 and P(A and B) = 0.4, what is P(A or B)? — 0.9
9. If P(A) = 0.5, P(B) = 0.4, and P(A|B) = 0.8, what is P(A and B)? — 0.32
10. Consider the statement: 1 + 2 + ... + n = n(n+1)/2. If the inductive step assumes P(k) is true, what needs to be proven? — 1 + 2 + ... + (k+1) = (k+1)(k+2)/2
11. What is the probability that a randomly selected integer from 1 to 10 is divisible by 3? — 3/10
12. What is the probability of drawing an ace from a standard deck of 52 cards? — 1/13
13. What is the probability of getting at least one head when flipping a fair coin 4 times? — 15/16
14. What is the probability of rolling a sum of 7 with two standard six-sided dice? — 6/36
15. What is the probability of drawing a spade or a heart from a standard deck of 52 cards? — 26/52
16. What is the probability of getting exactly 2 heads in 3 coin flips? — 3/8
17. Consider a set of 5 distinct books. How many ways can you arrange 3 of them on a shelf? — 60
18. In how many ways can a committee of 3 people be chosen from a group of 10 people? — 120
19. What is the minimum number of students in a class to guarantee that at least two students receive the same score on a test graded from 0 to 100? — 102
20. How many ways can the letters of the word 'BOOK' be arranged? — 12
21. How many ways can the letters of the word 'APPLE' be arranged? — 60
22. How many non-negative integer solutions are there to the equation x1 + x2 + x3 = 10? — 120
23. In a class of 30 students, 15 like Math and 20 like Science. If 5 students like both, how many students like either Math or Science? — 25
24. How many different outcomes are possible when flipping a coin 3 times? — 8
25. How many permutations are there of the set {a, b, c}? — 6
26. In a group of 20 people, 12 play cricket and 15 play football. If each person plays at least one game, how many play both? — 5
27. Suppose a bag contains 5 red balls and 3 blue balls. If you draw one ball at random, what is the probability it is red? — 5/8
28. How many distinct permutations are there of the letters in the word 'MISSISSIPPI'? — 11! / (4! * 4! * 2! * 1!)
29. What is the value of 5! (5 factorial)? — 120
30. In how many ways can 5 people be seated in a row? — 120
31. What is the total number of possible outcomes when rolling two distinct dice? — 36
32. If you have 3 shirts and 4 pants, how many different outfits can you create? — 12
33. How many ways can you choose a team of 4 players from a group of 7? — 28
34. What are the two steps involved in a proof by mathematical induction? — Base case and Inductive step
35. What is the principle of mathematical induction used for? — Proving statements about natural numbers
36. In probability, what is the set of all possible outcomes of an experiment called? — Sample Space
37. In Bayes' Theorem, P(A|B) = P(B|A) * P(A) / P(B), what is P(A) called? — Prior Probability
38. If a task can be performed in m ways, and after it is performed, a second task can be performed in n ways, then the two tasks can be performed in sequence in how many ways? — m * n
39. If a task can be performed in m ways and a second task can be performed in n ways, and these tasks cannot be performed at the same time, then the task can be performed in how many ways? — m + n
40. What is the formula for the number of permutations of n objects where there are n1 identical objects of type 1, n2 identical objects of type 2, ..., nk identical objects of type k? — n! / (n1! * n2! * ... * nk!)
41. What is the number of permutations of n distinct objects taken r at a time, denoted as P(n, r)? — n! / (n-r)!
42. What is the number of combinations of n distinct objects taken r at a time, denoted as C(n, r)? — n! / (r! * (n-r)!)
43. What is the formula for conditional probability P(A|B)? — P(A and B) / P(B)
44. If events A and B are independent, what is P(A and B)? — P(A) * P(B)
45. What is the fundamental principle that states if n items are put into m containers, with n > m, then at least one container must contain more than one item? — Pigeonhole Principle
46. Which principle is used to count the number of elements in the union of multiple sets? — Inclusion-Exclusion Principle
47. In the context of the Pigeonhole Principle, what are the 'pigeons' and 'pigeonholes' in the statement: 'If you have 5 pigeons and 4 pigeonholes, at least one pigeonhole must contain more than one pigeon'? — Pigeons are the items, pigeonholes are the containers.
48. If you are proving a statement P(n) for all integers n >= 1, what is the first step in mathematical induction? — Prove P(1) is true.
49. What is the base case in a proof by induction for the statement P(n)? — Proving P(1) is true
50. What does Bayes' Theorem describe? — The probability of an event given prior knowledge of related events
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