TRB PG Assistant - Probability - discrete sample space, events, classical, relative frequency and axiomatic approaches, conditional probability, independence, Bayes' theorem Online Test (MCQ Quiz)

1. What is a set of all possible outcomes of a random experiment called?

A) Event B) Sample Point C) Sample Space D) Probability Distribution

2. Which of the following represents a subset of the sample space?

A) Sample Space B) Event C) Outcome D) Experiment

3. In the classical approach to probability, what is the probability of an event E assumed to be?

A) Number of favorable outcomes / Total number of outcomes B) Limit of relative frequency as the number of trials increases C) Sum of probabilities of individual outcomes D) A number between 0 and 1, inclusive

4. When is the classical approach to probability applicable?

A) When outcomes are not equally likely B) When the number of outcomes is infinite C) When all outcomes are equally likely and finite D) When dealing with subjective beliefs

5. What does the relative frequency approach to probability estimate?

A) The inherent likelihood of an outcome before any trials B) The probability based on subjective judgment C) The probability of an event based on past occurrences D) The probability of an event in a theoretical setting

6. The relative frequency approach to probability is often used when:

A) The number of possible outcomes is small and equally likely B) It's impossible to determine equally likely outcomes C) The experiment can only be performed once D) The outcomes are deterministic

7. Which approach to probability is based on a set of axioms or postulates?

A) Classical Approach B) Relative Frequency Approach C) Axiomatic Approach D) Subjective Approach

8. In the axiomatic approach, what is the probability of any event P(E) guaranteed to be?

A) Greater than 1 B) Less than 0 C) Between 0 and 1, inclusive D) Exactly 1

9. According to the axiomatic approach, what is the probability of the sample space S?

A) 0 B) 0.5 C) 1 D) Undefined

10. If E1 and E2 are mutually exclusive events, what is P(E1 U E2) according to the axiomatic approach?

A) P(E1) + P(E2) - P(E1 ∩ E2) B) P(E1) * P(E2) C) P(E1) + P(E2) D) 1 - P(E1) - P(E2)

11. What is conditional probability denoted as P(A|B)?

A) Probability of A and B occurring together B) Probability of A occurring given that B has occurred C) Probability of A occurring or B occurring D) Probability of B occurring given that A has occurred

12. The formula for conditional probability P(A|B) is:

A) P(A ∩ B) / P(B) B) P(A ∩ B) / P(A) C) P(A) / P(A ∩ B) D) P(B) / P(A ∩ B)

13. For conditional probability P(A|B) to be defined, what must be true?

A) P(A) > 0 B) P(B) = 0 C) P(B) > 0 D) P(A ∩ B) = 0

14. What does it mean for two events A and B to be independent?

A) The occurrence of A affects the probability of B B) The occurrence of B affects the probability of A C) The occurrence of A does not affect the probability of B D) A and B cannot occur at the same time

15. If events A and B are independent, which of the following is true?

A) P(A ∩ B) = P(A) + P(B) B) P(A ∩ B) = P(A) / P(B) C) P(A ∩ B) = P(A) * P(B) D) P(A | B) = 1

16. If P(A|B) = P(A), what can be concluded about events A and B?

A) They are mutually exclusive B) They are dependent C) They are independent D) B is a subset of A

17. Bayes' theorem provides a way to:

A) Calculate the probability of an event before any evidence is considered B) Update the probability of a hypothesis based on new evidence C) Determine if two events are independent D) Calculate the probability of the union of two events

18. Bayes' theorem is mathematically expressed as:

A) P(A|B) = P(B|A)P(A) / P(B) B) P(A|B) = P(A ∩ B) / P(B) C) P(A U B) = P(A) + P(B) D) P(A) = P(A|B)P(B)

19. In Bayes' theorem, P(A|B) is called the:

A) Prior probability B) Likelihood C) Marginal probability D) Posterior probability

20. In Bayes' theorem, P(B|A) is often referred to as the:

A) Posterior probability B) Marginal probability C) Likelihood D) Prior probability

21. Consider rolling a fair six-sided die. What is the sample space?

A) {1, 2, 3, 4, 5, 6} B) {Even, Odd} C) {1, 3, 5} D) {2, 4, 6}

22. If you flip a fair coin twice, what is the sample space?

A) {HH, HT, TH, TT} B) {H, T} C) {HH, TT} D) {HT, TH}

23. Let S = {1, 2, 3, 4, 5, 6} be the sample space for rolling a die. What is the event of rolling an even number?

A) {1, 3, 5} B) {2, 4, 6} C) {1, 2, 3} D) {4, 5, 6}

24. What is the probability of drawing an ace from a standard deck of 52 cards in a single draw, using the classical approach?

A) 1/52 B) 4/52 C) 13/52 D) 1/13

25. A coin is tossed 100 times, and heads appears 45 times. What is the relative frequency estimate of the probability of getting heads?

A) 0.50 B) 0.45 C) 0.55 D) 100/45

26. Which axiom of probability states that probabilities are non-negative?

A) P(S) = 1 B) P(E) >= 0 for any event E C) For mutually exclusive events E_i, P(U E_i) = Sum P(E_i) D) P(E) = Number of favorable outcomes / Total outcomes

27. If P(A) = 0.6 and P(B) = 0.3, and A and B are independent, what is P(A ∩ B)?

A) 0.9 B) 0.3 C) 0.18 D) 0.0

28. If P(A) = 0.7 and P(B) = 0.4, and P(A ∩ B) = 0.2, what is P(A U B)?

A) 1.1 B) 0.9 C) 0.5 D) 0.7

29. If P(A) = 0.5 and P(B) = 0.6, and P(A ∩ B) = 0.4, what is P(A|B)?

A) 0.5 / 0.6 B) 0.4 / 0.6 C) 0.4 / 0.5 D) 0.5 / 0.4

30. Let A be the event of getting a sum of 7 when rolling two dice, and B be the event of getting a 4 on the first die. Are A and B independent?

A) Yes, because P(A ∩ B) = P(A) * P(B) B) No, because P(A ∩ B) != P(A) * P(B) C) Yes, because P(A|B) = P(A) D) No, because P(A|B) != P(A)

31. Suppose a medical test for a disease is 99% accurate (detects the disease if present) and has a 1% false positive rate (indicates disease if not present). If 0.5% of the population has the disease, what is the probability that a person has the disease given a positive test result?

A) 0.99 B) 0.01 C) Approximately 0.99 D) Approximately 0.33

32. Which of the following is NOT a property of probability?

A) Probability of an event is between 0 and 1. B) Probability of the sample space is 1. C) Probability of the union of mutually exclusive events is the sum of their probabilities. D) Probability of an impossible event is 1.

33. In a discrete sample space, what is the probability of an event E that is composed of a finite number of sample points {e1, e2, ..., ek}?

A) P(E) = P(e1) + P(e2) + ... + P(ek) B) P(E) = P(e1) * P(e2) * ... * P(ek) C) P(E) = (Number of points in E) / (Total number of points in S) D) P(E) = 1 - P(E')

34. Consider the experiment of selecting a letter from the word 'STATISTICS'. What is the probability of selecting the letter 'S'?

A) 3/10 B) 4/10 C) 1/10 D) 2/10

35. What is the probability of drawing a red card or a face card (King, Queen, Jack) from a standard deck of 52 cards?

A) 26/52 + 12/52 B) 26/52 + 12/52 - 6/52 C) 26/52 + 12/52 - 8/52 D) 26/52 + 12/52 - 4/52

36. If P(A) = 0.4, P(B) = 0.5, and P(A U B) = 0.7, are events A and B independent?

A) Yes, because P(A) + P(B) = 0.9 != 0.7 B) No, because P(A ∩ B) = P(A) + P(B) - P(A U B) = 0.4 + 0.5 - 0.7 = 0.2, and P(A) * P(B) = 0.4 * 0.5 = 0.2 C) No, because P(A ∩ B) = 0.2, and P(A) * P(B) = 0.2 D) Yes, because P(A ∩ B) = 0.2, and P(A) * P(B) = 0.2

37. What is the probability of event A occurring given that event B has NOT occurred, P(A|B')?

A) P(A ∩ B') / P(B') B) P(A ∩ B) / P(B') C) P(A U B') / P(B') D) P(A) / P(B')

38. If P(A) = 0.3, P(B) = 0.6, and P(A|B) = 0.2, what is P(A ∩ B)?

A) 0.3 * 0.6 B) 0.2 * 0.6 C) 0.3 * 0.2 D) 0.6 / 0.2

39. In Bayes' theorem, P(A) is referred to as the:

A) Posterior probability B) Likelihood C) Prior probability D) Conditional probability

40. What is the probability of rolling a sum of 8 with two fair dice?

A) 5/36 B) 6/36 C) 4/36 D) 3/36

41. If events A and B are mutually exclusive, what is P(A ∩ B)?

A) P(A) * P(B) B) P(A) + P(B) C) 0 D) 1

42. What is the probability of selecting a vowel from the letters of the word 'EQUATION'?

A) 5/8 B) 4/8 C) 3/8 D) 6/8

43. If P(A) = 0.6, P(A') = 0.4. If P(B|A) = 0.5, what is P(A|B) if P(B) = 0.7?

A) (0.6 * 0.5) / 0.7 B) (0.4 * 0.5) / 0.7 C) (0.5 * 0.7) / 0.6 D) (0.6 * 0.7) / 0.5

44. Which approach to probability is most suitable for situations involving personal beliefs or opinions?

A) Classical Approach B) Relative Frequency Approach C) Axiomatic Approach D) Subjective Approach (though not explicitly detailed in the topic, it's a contrasting concept)

45. Let S = {1, 2, 3, ..., 10}. Let E be the event that a number chosen from S is divisible by 3. What is P(E)?

A) 3/10 B) 4/10 C) 2/10 D) 1/10

46. If P(A) = 0.8 and P(B) = 0.7, what is the minimum possible value for P(A ∩ B)?

A) 0.0 B) 0.1 C) 0.5 D) 0.8

47. Consider the events A: 'It is raining' and B: 'The ground is wet'. If P(A) = 0.3, P(B) = 0.4, and P(B|A) = 0.8, what is P(A ∩ B)?

A) 0.3 * 0.8 B) 0.4 * 0.8 C) 0.3 * 0.4 D) 0.8 / 0.3

48. What is the probability that a randomly selected integer from 1 to 30 is divisible by either 4 or 6?

A) 12/30 B) 10/30 C) 8/30 D) 9/30

49. If two events A and B are independent, then A' and B' are also:

A) Dependent B) Independent C) Mutually Exclusive D) Complementary

Probability - discrete sample space, events, classical, relative frequency and axiomatic approaches, conditional probability, independence, Bayes' theorem - Online Test

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