وَأَحْصَىٰ كُلَّ شَيْءٍ عَدَدًا
"And He has enumerated everything in numbers." (Allah perfectly counts and knows the exact measure of all creations).
Q1. A set consisting of all elements of the sets
under consideration is called:
A. Universal set
B. Sample space
C. Sub set
D. Both (a) and (b) ✓ Correct Answer
Explanation: In probability, the universal set (containing all
elements under consideration) is the same concept as the sample space.
Q2. The set of all possible outcomes of a random
experiment is called:
A. Universal set
B. Sample space ✓ Correct Answer
C. Sub set
D. Both (a) and (b)
Explanation: By definition, the sample space is the set of every
possible outcome of a random experiment.
Q3. Any subset of the sample space is called:
A. Element
B. Sample point
C. Impossible event
D. Event ✓ Correct Answer
Explanation: In probability theory, an event is defined as any subset
of the sample space.
Q4. A set containing only one element is called a:
A. Simple event ✓ Correct Answer
B. Compound event
C. Empty set
D. Null event
Explanation: An event with exactly one sample point is called a simple
event.
Q5. A set containing more than one element is called:
A. Simple event
B. Compound event ✓ Correct Answer
C. Empty set
D. Null event
Explanation: An event made up of two or more sample points is called a
compound event.
Q6. A set containing no element of the sample space
is called:
A. Impossible event
B. Null set
C. Empty set
D. All of these ✓ Correct Answer
Explanation: An event with no outcomes is called the impossible event,
the null set, or the empty set — all describe the same thing.
Q7. A set containing all elements of the sample space
is called:
A. Impossible event
B. Null set
C. Empty set
D. Sure event ✓ Correct Answer
Explanation: An event that includes every outcome in the sample space
is called the sure (certain) event.
Q8. If A∩B = ∅, then A and B are:
A. Disjoint
B. Mutually exclusive
C. Both (a) and (b) ✓ Correct Answer
D. Neither
Explanation: An empty intersection means A and B share no common
outcomes — they are both disjoint and mutually exclusive.
Q9. Two events A and B are called mutually exclusive
if:
A. They are disjoint
B. A∩B = ∅
C. P(A∩B) = 0
D. All of these ✓ Correct Answer
Explanation: Being disjoint, having an empty intersection, and having
zero joint probability are all equivalent descriptions of mutually exclusive
events.
Q10. Two events that cannot occur at the same time
are called:
A. Mutually exclusive ✓ Correct Answer
B. Not mutually exclusive
C. Overlapping
D. None
Explanation: By definition, mutually exclusive events cannot both
happen simultaneously.
Q11. A and B cannot occur together means they are:
A. Not mutually exclusive
B. Disjoint ✓ Correct Answer
C. Overlapping
D. Both (a) and (c)
Explanation: Events that cannot occur together are, by definition,
disjoint.
Q12. In the tossing of a coin, either the head or the
tail will appear — these events are:
A. Not mutually exclusive
B. Disjoint
C. Mutually exclusive
D. Both (b) and (c) ✓ Correct Answer
Explanation: A single coin toss gives either heads or tails, never
both — so these outcomes are disjoint and mutually exclusive.
Q13. If A∩B ≠ ∅, then A and B are:
A. Overlapping
B. Not mutually exclusive
C. Mutually exclusive
D. Both (a) and (b) ✓ Correct Answer
Explanation: A non-empty intersection means the events share outcomes
— making them overlapping and not mutually exclusive.
Q14. Two events that are exhaustive but always
mutually exclusive are called:
A. Complementary events ✓ Correct Answer
B. Equally likely events
C. Empty set
D. Null event
Explanation: Complementary events (like A and its complement) together
cover the whole sample space, yet never overlap.
Q15. Two events having no common point are called:
A. Disjoint events
B. Mutually exclusive events
C. Not mutually exclusive events
D. Both (a) and (b) ✓ Correct Answer
Explanation: Sharing no common outcome describes both disjoint and
mutually exclusive events — they are the same idea.
Q16. If A∪B = S, then A and B are always:
A. Exhaustive events ✓ Correct Answer
B. Mutually exclusive events
C. Not mutually exclusive events
D. None
Explanation: When the union of two events covers the entire sample
space, they are called exhaustive events.
Q17. Events whose union is the sample space are
called:
A. Exhaustive events
B. Complementary events
C. Mutually exclusive
D. Both (a) and (b) ✓ Correct Answer
Explanation: Both exhaustive events and complementary events share the
property that their union equals the sample space.
Q18. Two complementary events are always:
A. Mutually exclusive
B. Not mutually exclusive
C. Exhaustive
D. Both (a) and (c) ✓ Correct Answer
Explanation: Complementary events never overlap (mutually exclusive)
and together cover the whole sample space (exhaustive).
Q19. If n(A) = n(B), then A and B are:
A. Equally likely
events ✓ Correct Answer
B. Mutually exclusive events
C. Not mutually exclusive events
D. None
Explanation: Events with the same number of favourable outcomes have
equal probability, making them equally likely.
Q20. Equally likely events have:
A. Same chances ✓ Correct Answer
B. Different chances
C. Zero chances
D. Negative probability
Explanation: By definition, equally likely events have exactly the
same chance of occurring.
Q21. Events that have the same chance of occurrence
always are called:
A. Not mutually exclusive
B. Disjoint
C. Mutually exclusive
D. Equally likely ✓ Correct Answer
Explanation: Equal chances of occurrence is precisely what defines
equally likely events.
Q22. A set containing a specific (fixed) number of
elements is called a:
A. Finite set ✓ Correct Answer
B. Infinite set
C. Uncountable set
D. None
Explanation: A set with a definite, countable number of elements is
called a finite set.
Q23. The possible outcomes of a random experiment
must be:
A. At least one
B. At least two ✓ Correct Answer
C. Only one
D. None
Explanation: For an experiment to be considered 'random' with genuine
uncertainty, there must be at least two possible outcomes.
Q24. If the occurrence of one event doesn't affect
the chances of another, the events are:
A. Independent events ✓ Correct Answer
B. Dependent events
C. Random events
D. None
Explanation: This is the defining property of independent events.
Q25. If the occurrence of one event affects the
chances of another, the events are:
A. Independent events
B. Dependent events ✓ Correct Answer
C. Random events
D. None
Explanation: When one event's outcome changes the probability of
another, they are dependent events.
Q26. The life and death of two unrelated persons are:
A. Dependent
B. Independent ✓ Correct Answer
C. Mutually exclusive
D. None
Explanation: One person's life/death has no bearing on another
unrelated person's, making these independent events.
Q27. Drawing two cards without replacement gives
events that are:
A. Dependent ✓ Correct Answer
B. Independent
C. Both (a) and (b)
D. None
Explanation: Without replacement, the outcome of the first draw
changes the composition of the deck, affecting the second draw — dependent
events.
Q28. Drawing two cards with replacement gives events
that are:
A. Dependent
B. Independent ✓ Correct Answer
C. Both (a) and (b)
D. None
Explanation: With replacement, the deck returns to its original state
before the second draw, so the draws are independent.
Q29. An arrangement of objects in a definite order is
called a:
A. Permutation ✓ Correct Answer
B. Combination
C. Set
D. None
Explanation: A permutation is an arrangement where the order of
objects matters.
Q30. The number of ways of arranging n objects taken
r at a time in a definite order is:
A. ⁿCᵣ
B. ⁿPᵣ ✓ Correct Answer
C. n!/[r!(n-r)!]
D. None
Explanation: The permutation formula, ⁿPᵣ, counts ordered arrangements
of r objects from n.
Q31. An arrangement of objects without a definite
order is called a:
A. Permutation
B. Combination ✓ Correct Answer
C. Ordered pair
D. None
Explanation: A combination is a selection of objects where order does
not matter.
Q32. The number of ways of arranging n objects taken
r at a time without any order is:
A. ⁿCᵣ ✓ Correct Answer
B. ⁿPᵣ
C. r-tuple
D. None
Explanation: The combination formula, ⁿCᵣ, counts unordered selections
of r objects from n.
Q33. The orderly arrangements of r distinct things
out of n are called:
A. Permutations
B. Combinations
C. r-tuples
D. Both (a) and (c) ✓ Correct Answer
Explanation: An ordered arrangement of r items is called both a
permutation and, formally, an r-tuple.
Q34. A non-orderly arrangement of things is called a:
A. Permutation
B. Combination ✓ Correct Answer
C. r-tuple
D. None
Explanation: When order doesn't matter, the arrangement is called a
combination.
Q35. The number of permutations of n objects, of
which n₁ are alike, n₂ are alike, and so on, is:
A. n! / (n₁! × n₂! × ... ×
nₖ!) ✓ Correct Answer
B. ⁿCᵣ
C. ⁿPᵣ
D. None
Explanation: This is the standard formula for permutations of objects
with repeated (indistinguishable) items.
Q36. A coin is tossed three times; the total number
of sample points will be:
A. 2³ ✓ Correct Answer
B. 2²
C. 3²
D. None
Explanation: Each toss has 2 outcomes, and with 3 tosses the total
sample points = 2×2×2 = 2³ = 8.
Q37. In the tossing of a coin and a die together, the
total number of sample points is:
A. 12 ✓ Correct Answer
B. 8
C. 4
D. None
Explanation: The coin has 2 outcomes and the die has 6, so total
sample points = 2 × 6 = 12.
Q38. 0! =
A. Undefined
B. 1 ✓ Correct Answer
C. 0
D. None
Explanation: By convention, 0! is defined to equal 1.
Q39. It is necessary to perform the experiment a
large number of times in the:
A. Classical approach
B. Axiomatic approach
C. Posterior (relative
frequency) approach ✓ Correct Answer
D. None
Explanation: The posterior/relative-frequency approach to probability
estimates likelihood by repeating an experiment many times.
Q40. The probability of a sure event always equals:
A. 1 ✓ Correct Answer
B. Zero
C. Negative
D. Undefined
Explanation: A sure (certain) event always has probability exactly
equal to 1.
Q41. The probability of the sample space is equal to:
A. 1 ✓ Correct Answer
B. Zero
C. Negative
D. Undefined
Explanation: Since the sample space includes every possible outcome,
its total probability is always 1.
Q42. The probability of an event can be any number
between and including:
A. 0 and 1 ✓ Correct Answer
B. -1 and 0
C. -1 and +1
D. None
Explanation: Probability values always lie in the closed interval [0,
1].
Q43. If an event can't occur, its probability is:
A. Negative
B. Positive
C. Zero ✓ Correct Answer
D. None
Explanation: An impossible event (one that cannot occur) has a
probability of exactly zero.
Q44. The probability of an event can't be:
A. Greater than one
B. Less than zero
C. Negative
D. All of these ✓ Correct Answer
Explanation: Probability is bounded between 0 and 1, so it can never
be negative or exceed one — ruling out all these options.
Q45. The probability of an impossible event is:
A. 1
B. 0 ✓ Correct Answer
C. Less than zero
D. None
Explanation: By definition, an event that can never happen has a
probability of 0.
Q46. The sum of probabilities of all events in the
sample space is:
A. 1 ✓ Correct Answer
B. Greater than 1
C. Less than 1
D. -1
Explanation: The total probability across the entire sample space must
always sum to exactly 1.
Q47. If you bought 7 tickets out of 700 tickets, your
probability of winning is:
A. 7/700 ✓ Correct Answer
B. 700/7
C. 700 + 7
D. 693/700
Explanation: Probability of winning = (favourable outcomes)/(total
outcomes) = 7/700.
Q48. The probability of drawing a spade card from a
pack of 52 cards is:
A. 12/52
B. 13/52 ✓ Correct Answer
C. 26/52
D. 52/13
Explanation: There are 13 spade cards in a deck of 52, so the
probability is 13/52.
Q49. The probability of throwing an 8 with a single
six-sided die is:
A. 1
B. -1
C. Zero ✓ Correct Answer
D. None
Explanation: A standard die only shows 1 through 6, so getting an 8 is
impossible — probability 0.
Q50. A single letter is selected from the word
'probability'; what is the probability it is a vowel?
A. 4/11 ✓ Correct Answer
B. 6/11
C. Zero
D. None
Explanation: 'Probability' has 11 letters, of which o, a, i, i are
vowels — 4 vowels out of 11 letters.
Q51. The probability of drawing a pictured (face)
card from a pack of 52 cards is:
A. 12/52 ✓ Correct Answer
B. 26/52
C. 4/52
D. 52/12
Explanation: There are 12 face cards (Jack, Queen, King in each of 4
suits) out of 52 cards.
Q52. The probability of drawing a white ball from a
bag containing red, black, green and white balls, where there are 5 white balls
out of 11 total, is:
A. 11/11
B. 5/11 ✓ Correct Answer
C. 8/11
D. 10/11
Explanation: Probability = favourable (white) / total balls = 5/11.
Q53. The probability of getting an even number when a
tetrahedral (4-faced) die is thrown is:
A. 1/2 ✓ Correct Answer
B. 1/4
C. 3/4
D. 4/4
Explanation: A tetrahedral die has faces 1-4; the even numbers (2, 4)
make up 2 out of 4 faces, giving 1/2.
Q54. In tossing a six-sided die, each face has
probability:
A. 4/6
B. 3/6
C. 2/6
D. 1/6 ✓ Correct Answer
Explanation: A fair six-sided die gives each of its 6 faces an equal
probability of 1/6.
Q55. Two events A and B are mutually exclusive if:
A. A∩B = ∅ ✓ Correct Answer
B. A∩B ≠ ∅
C. P(A∩B) > 0
D. None
Explanation: Mutually exclusive events, by definition, have an empty
intersection.
Q56. In the tossing of two coins, the probability of
getting exactly one head is:
A. -1/2
B. 1/2 ✓ Correct Answer
C. 0
D. None
Explanation: Out of 4 equally likely outcomes (HH, HT, TH, TT),
exactly 2 have one head, giving probability 2/4 = 1/2.
Q57. Which of the following is a valid probability
value for drawing a ball from a bag?
A. -1/2
B. 3/2
C. 3/5 ✓ Correct Answer
D. 2
Explanation: Since probability must lie between 0 and 1, only 3/5 is a
valid probability value among these options.
Q58. If P(A∩B) = 0, then A and B are:
A. Disjoint
B. Mutually exclusive events
C. Both (a) and (b) ✓ Correct Answer
D. Not mutually exclusive
Explanation: A zero joint probability means A and B share no outcomes
— they are both disjoint and mutually exclusive.
Q59. n! denotes the:
A. Product of n
integers ✓ Correct Answer
B. Addition of n integers
C. Subtraction of n integers
D. Division of n integers
Explanation: Factorial notation, n!, represents the product of all
positive integers from 1 up to n.
Q60. If P(A∪B) = 1, then A and B would be:
A. Exhaustive ✓ Correct Answer
B. Complementary
C. Mutually exclusive
D. Both (a) and (b)
Explanation: If the union of two events covers the whole sample space
(probability 1), they are exhaustive events.
Q61. If A ⊂ S, then P(A) + P(Ā) =
A. Zero
B. 1 ✓ Correct Answer
C. Less than 1
D. Greater than 1
Explanation: An event and its complement always cover the entire sample
space, so their probabilities always sum to 1.
Q62. If two cards are drawn from 52 cards, the total
number of possible outcomes is:
A. 52
B. 1326 ✓ Correct Answer
C. 100
D. Undefined
Explanation: The number of ways to choose 2 cards out of 52 (unordered)
is C(52,2) = 1326.
Q63. If A and B are two non-overlapping (mutually
exclusive) events, then P(A∪B) =
A. P(A)·P(B)
B. P(A) + P(B) ✓ Correct Answer
C. P(A) + P(B) - P(A∩B)
D. -0.1
Explanation: For mutually exclusive events, since P(A∩B)=0, the
addition rule simplifies to P(A) + P(B).
Q64. If A and B are two overlapping (not mutually
exclusive) events, then P(A∪B) =
A. P(A)·P(B)
B. P(A) + P(B)
C. P(A) + P(B) - P(A∩B) ✓ Correct Answer
D. -0.1
Explanation: For overlapping events, the general addition rule
requires subtracting the shared probability: P(A)+P(B)-P(A∩B).
Q65. If A and B are two dependent (not independent)
events, then P(A∩B) =
A. P(A)·P(B)
B. P(A) + P(B)
C. P(A) + P(B) - P(A∩B)
D. P(A)·P(B|A) ✓ Correct Answer
Explanation: For dependent events, the joint probability uses
conditional probability: P(A∩B) = P(A)·P(B|A).
Q66. If A and B are independent events, P(A) = 0.25,
P(B) = 0.40, then P(A∩B) =
A. 0.1 ✓ Correct Answer
B. 0.65
C. 0.625
D. -0.1
Explanation: For independent events, P(A∩B) = P(A)×P(B) = 0.25 × 0.40
= 0.1.
Q67. If P(A∩B) = 0, then A and B are:
A. Disjoint
B. Mutually exclusive events
C. Both (a) and (b) ✓ Correct Answer
D. Not mutually exclusive
Explanation: Zero joint probability again confirms A and B are both
disjoint and mutually exclusive.
Q68. The probability of an event cannot be:
A. Greater than 0
B. Less than 0 ✓ Correct Answer
C. Equal to 1
D. Equal to 0
Explanation: Probability values can never be negative — that's the one
option that's never valid.
Q69. If two coins are tossed, the probability of
getting one head and one tail is:
A. 1/4
B. 2/4 ✓ Correct Answer
C. 3/4
D. 2/3
Explanation: Out of 4 equally likely outcomes (HH, HT, TH, TT), 2 give
one head and one tail, so the probability is 2/4.
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