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Unit 1: Introduction to Probability - Practice Quiz

ECAP790 60 Questions
0 Correct 0 Wrong 60 Left
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1 What does the union represent?

Elements of Set Theory Easy
A. Elements in , in , or in both
B. Elements common to both and
C. Elements outside both and
D. Elements in but not in

2 What does the intersection represent?

Elements of Set Theory Easy
A. Elements belonging to neither set
B. Elements belonging to both and
C. Elements belonging only to set
D. Elements belonging only to set

3 If the universal set is and , what does denote?

Elements of Set Theory Easy
A. The elements in that are not in
B. The elements in that are not in
C. The elements in both and
D. The elements shared by and

4 What is a sample space in a probability experiment?

Sample Space and Probability Measure Easy
A. The set of all possible outcomes
B. The set of favorable outcomes only
C. The numerical probability of an event
D. The outcome occurring most often

5 What is the sample space when a coin is tossed once?

Sample Space and Probability Measure Easy
A.
B.
C.
D.

6 For any sample space , what is ?

Sample Space and Probability Measure Easy
A.
B.
C.
D.

7 Events and are independent when which equality holds?

Statistical Independence Easy
A.
B.
C.
D.

8 If two events are independent, what does the occurrence of one event do to the probability of the other?

Statistical Independence Easy
A. It leaves the probability unchanged
B. It makes the probability equal to zero
C. It always decreases the probability
D. It always increases the probability

9 Which formula defines when ?

Conditional Probability Easy
A.
B.
C.
D.

10 What does mean?

Conditional Probability Easy
A. The probability of given that occurred
B. The probability that neither event occurred
C. The probability that exactly one event occurred
D. The probability of given that occurred

11 If and , what is ?

Conditional Probability Easy
A.
B.
C.
D.

12 A coin is tossed twice. How many sample points are in the sample space?

Counting Sample Points Easy
A.
B.
C.
D.

13 A six-sided die is rolled and a coin is tossed. How many possible combined outcomes are there?

Counting Sample Points Easy
A.
B.
C.
D.

14 How many different arrangements can be made using the three distinct letters A, B, and C exactly once?

Counting Sample Points Easy
A.
B.
C.
D.

15 Three events are pairwise independent if which condition holds?

Mutually and pairwise independent events Easy
A. All three events are mutually exclusive
B. All three events have equal probabilities
C. Every pair of events is independent
D. Exactly one pair of events is independent

16 For three events , , and to be mutually independent, which additional equality is required beyond pairwise independence?

Mutually and pairwise independent events Easy
A.
B.
C.
D.

17 If independent events and have probabilities and , what is ?

Multiplication theorem of probability for independent events Easy
A.
B.
C.
D.

18 If three independent events each have probability , what is the probability that all three occur?

Multiplication theorem of probability for independent events Easy
A.
B.
C.
D.

19 Which formula is Bayes' theorem for events and with positive probabilities?

Bayes’ theorem Easy
A.
B.
C.
D.

20 In Bayes' theorem, what is commonly called?

Bayes’ theorem Easy
A. The joint probability
B. The posterior probability
C. The conditional likelihood
D. The prior probability

21 Let , with the multiples of and the multiples of . How many elements are in ?

Elements of Set Theory Medium
A.
B.
C.
D.

22 For finite sets , , and , suppose , , , , , , and . Find .

Elements of Set Theory Medium
A.
B.
C.
D.

23 If , , and , what is ?

Elements of Set Theory Medium
A.
B.
C.
D.

24 A random variable takes values in , and the probability of outcome is . What is the probability of obtaining an even outcome?

Sample Space and Probability Measure Medium
A.
B.
C.
D.

25 For events and , suppose , , and . What is the probability that exactly one of the events occurs?

Sample Space and Probability Measure Medium
A.
B.
C.
D.

26 How many five-digit numbers can be formed from the digits if no digit may be repeated?

Counting Sample Points Medium
A.
B.
C.
D.

27 A committee of is selected from men and women. How many committees contain at least women?

Counting Sample Points Medium
A.
B.
C.
D.

28 A card is selected from a standard -card deck. Given that the card is red, what is the probability that it is a face card?

Conditional Probability Medium
A.
B.
C.
D.

29 Two fair dice are rolled. Given that their sum is greater than , what is the probability that at least one die shows ?

Conditional Probability Medium
A.
B.
C.
D.

30 Suppose , , and . Which statement is correct?

Statistical Independence Medium
A. and are dependent, and .
B. and are independent, and .
C. and are independent, and .
D. and are dependent, and .

31 Suppose , , and . What is , and are and independent?

Statistical Independence Medium
A. , and they are independent
B. , and they are dependent
C. , and they are independent
D. , and they are dependent

32 Which statement about three events , , and is always true?

Mutually and pairwise independent events Medium
A. Mutual independence implies equal event probabilities.
B. Mutual independence implies pairwise independence.
C. Pairwise independence implies mutual independence.
D. Pairwise independence implies the events are disjoint.

33 Two fair coins are tossed. Let be the event that the first coin is heads, that the second coin is heads, and that both coins show the same result. How are , , and related?

Mutually and pairwise independent events Medium
A. They are mutually independent and pairwise independent.
B. They are mutually independent but not pairwise independent.
C. They are pairwise independent but not mutually independent.
D. They are neither pairwise nor mutually independent.

34 Events , , and satisfy , every pairwise intersection has probability , and . Which conclusion follows?

Mutually and pairwise independent events Medium
A. The events are conditionally dependent.
B. The events are pairwise independent only.
C. The events are mutually independent.
D. The events are mutually exclusive.

35 Three independent components operate successfully with probabilities , , and . What is the probability that all three operate successfully?

Multiplication theorem of probability for independent events Medium
A.
B.
C.
D.

36 Three independent experiments have success probabilities , , and . What is the probability of at least one success?

Multiplication theorem of probability for independent events Medium
A.
B.
C.
D.

37 Factory produces of a company's items and has a defect rate of . Factory produces and has a defect rate of . Given that an item is defective, what is the probability it came from factory ?

Bayes’ theorem Medium
A.
B.
C.
D.

38 A condition affects of a population. A test has sensitivity and specificity . If a person tests positive, what is the probability that the person has the condition?

Bayes’ theorem Medium
A.
B.
C.
D.

39 One of two urns is selected with equal probability. Urn contains red and blue ball, while urn contains red and blue balls. A selected ball is red. What is the probability that urn was chosen?

Bayes’ theorem Medium
A.
B.
C.
D.

40 A code consists of three letters followed by three digits. Repetition is allowed. How many codes contain at least one even digit, where is considered even?

Counting Sample Points Medium
A.
B.
C.
D.

41 Events , , and satisfy , , , , , , and . What is the probability that exactly one of the events occurs?

Elements of Set Theory Hard
A.
B.
C.
D.

42 Let and define , , and . How many sets belong to the -algebra ?

Elements of Set Theory Hard
A.
B.
C.
D.

43 On , suppose the probability measure satisfies for . What is ?

Sample Space and Probability Measure Hard
A.
B.
C.
D.

44 Let in . An assignment claims that for every . Which probability-measure property shows that this assignment is impossible?

Sample Space and Probability Measure Hard
A. Continuity from below for probability measures
B. Continuity from above for probability measures
C. Monotonicity under event-set inclusion
D. Finite additivity on disjoint event pairs

45 Independent events and satisfy and . What are and ?

Statistical Independence Hard
A.
B.
C.
D.

46 Independent events and satisfy and , where . Find .

Statistical Independence Hard
A.
B.
C.
D.

47 Suppose , , and . What is ?

Conditional Probability Hard
A.
B.
C.
D.

48 Events and satisfy , , and . Find .

Conditional Probability Hard
A.
B.
C.
D.

49 How many distinct arrangements of the letters in MISSISSIPPI contain no two occurrences of the letter I adjacent?

Counting Sample Points Hard
A.
B.
C.
D.

50 How many five-card hands from a standard 52-card deck contain cards from exactly two suits, with both selected suits represented?

Counting Sample Points Hard
A.
B.
C.
D.

51 How many onto functions from a seven-element labeled set to a four-element labeled set have exactly one codomain element with a preimage of size four?

Counting Sample Points Hard
A.
B.
C.
D.

52 Two fair coins are tossed. Let be the event that the first coin is heads, that the second coin is heads, and that exactly one coin is heads. Which statement is correct?

Mutually and pairwise independent events Hard
A. The three events are mutually independent
B. No pair among the three events is independent
C. They are pairwise but not mutually independent
D. Only the events and are independent

53 Events , , and are pairwise independent and each has probability . What is the complete possible range of ?

Mutually and pairwise independent events Hard
A.
B.
C.
D.

54 Events , , and are pairwise independent, with . What is the smallest possible value of ?

Mutually and pairwise independent events Hard
A.
B.
C.
D.

55 A system has four mutually independent components with working probabilities , , , and . The system works if at least three components work. What is its reliability?

Multiplication theorem of probability for independent events Hard
A.
B.
C.
D.

56 Let be mutually independent events with . If , what is the probability that exactly one of these events occurs?

Multiplication theorem of probability for independent events Hard
A.
B.
C.
D.

57 A disease has prevalence . A test has sensitivity and specificity . Two test outcomes are conditionally independent given disease status. What is the probability of disease after two positive results?

Bayes’ theorem Hard
A.
B.
C.
D.

58 A coin is selected with equal prior probability from a fair coin and a coin whose head probability is . After observing the sequence H, T, H, what is the predictive probability that the next toss is H?

Bayes’ theorem Hard
A.
B.
C.
D.

59 One of three boxes is selected uniformly. Box A contains four red and two blue balls, Box B contains three red and three blue balls, and Box C contains two red and four blue balls. Two balls are drawn without replacement and appear red then blue. What is the posterior probability that Box B was selected?

Bayes’ theorem Hard
A.
B.
C.
D.

60 A positive-integer-valued random variable has for . For a positive integer , what is ?

Sample Space and Probability Measure Hard
A.
B.
C.
D.
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Unit 2