وَأَحْصَىٰ كُلَّ شَيْءٍ عَدَدًا
"And He has enumerated everything in numbers." (Allah perfectly counts and knows the exact measure of all creations).
Q1. A Bernoulli trial has only:
A. Two outcomes ✓ Correct Answer
B. Three outcomes
C. Four outcomes
D. Unlimited outcomes
Explanation: A Bernoulli trial is defined as a random experiment with
exactly two possible outcomes — success or failure.
Q2. If a Bernoulli trial is repeated independently n
> 1 times, it becomes a(n):
A. Hypergeometric experiment
B. Undefined experiment
C. Binomial experiment ✓ Correct Answer
D. Dependent experiment
Explanation: Repeating independent Bernoulli trials n times, with a
fixed probability of success, defines a binomial experiment.
Q3. If n = 1 in a Binomial distribution, then it
becomes a:
A. Uniform Distribution
B. Hypergeometric Distribution
C. Bernoulli
Distribution ✓ Correct Answer
D. None
Explanation: A single binomial trial (n=1) is exactly the same as a
single Bernoulli trial.
Q4. Throwing a coin once is an example of a:
A. Binomial experiment
B. Bernoulli trial ✓ Correct Answer
C. Hypergeometric experiment
D. None
Explanation: A single coin toss has only two outcomes (heads/tails),
making it a Bernoulli trial rather than a full binomial experiment (which needs
repeated trials).
Q5. A paper has 15 multiple choice questions with 3
alternatives; answering these by guesswork follows a:
A. Binomial
Distribution ✓ Correct Answer
B. Uniform Distribution
C. Hypergeometric Distribution
D. None
Explanation: Each question is an independent trial with a fixed
probability of guessing correctly, matching the binomial setup.
Q6. A random sample chosen with replacement from a
finite population follows a:
A. Dependent sample
B. Binomial
Distribution ✓ Correct Answer
C. Hypergeometric Distribution
D. None
Explanation: Sampling with replacement keeps trials independent with
constant probability, which is exactly the binomial setup.
Q7. Successive trials being independent is
characteristic of:
A. Binomial Distribution
B. Selection with replacement
C. Hypergeometric Distribution
D. Both (a) and (b) ✓ Correct Answer
Explanation: Both the binomial distribution and sampling with
replacement rely on independent successive trials.
Q8. The probability of success remaining constant at
each trial is characteristic of:
A. Hypergeometric Distribution
B. Dependent trials
C. Binomial Distribution ✓ Correct Answer
D. Selection without replacement
Explanation: A defining feature of the binomial distribution is that
the probability of success stays fixed across all trials.
Q9. The Binomial distribution is applicable when
successive trials are:
A. Done with replacement
B. Independent
C. Dependent
D. Both (a) and (b) ✓ Correct Answer
Explanation: Binomial trials require both independence and (typically
achieved via) sampling with replacement.
Q10. Tossing a fair coin 8 times is an example of a:
A. Binomial experiment ✓ Correct Answer
B. Uniform experiment
C. Hypergeometric experiment
D. None
Explanation: Each toss is an independent trial with a fixed
probability of heads, making repeated tosses a binomial experiment.
Q11. Drawing two cards with replacement is an example
of a:
A. Uniform experiment
B. Binomial experiment ✓ Correct Answer
C. Hypergeometric experiment
D. None
Explanation: Replacing the card after each draw keeps trials
independent and probabilities fixed — a binomial setup.
Q12. In a Binomial distribution, n (the number of
trials) is:
A. Fixed ✓ Correct Answer
B. Infinite
C. Unlimited
D. Not fixed
Explanation: The number of trials, n, is always a fixed,
pre-determined value in a binomial distribution.
Q13. In a Binomial Distribution, each trial has:
A. Four outcomes
B. Three outcomes
C. Two outcomes ✓ Correct Answer
D. Unlimited outcomes
Explanation: Each trial in a binomial setting is Bernoulli-like,
having exactly two possible outcomes.
Q14. The type of Binomial Distribution is:
A. Mixed
B. Continuous
C. Discrete ✓ Correct Answer
D. None
Explanation: Since the binomial random variable counts whole-number
successes, it is a discrete distribution.
Q15. A binomial random variable (with n trials) can
take:
A. n+1 possible values ✓ Correct Answer
B. n possible values
C. n-1 possible values
D. None
Explanation: A binomial variable can take values 0, 1, 2, ..., n —
that's n+1 distinct possible values.
Q16. In a Binomial Distribution, the total number of
successes in n trials is at most:
A. -n
B. n ✓ Correct Answer
C. n-1
D. n+1
Explanation: The maximum possible number of successes in n trials is n
itself.
Q17. In a Binomial Distribution, (q + p)ⁿ =
A. 0
B. -1
C. 1 ✓ Correct Answer
D. Zero
Explanation: Since p + q = 1 (probabilities of success and failure sum
to 1), raising it to any power n still gives 1.
Q18. In a Binomial Distribution, np(q + p)ⁿ =
A. np(q+p)ⁿ⁺¹
B. np ✓ Correct Answer
C. 1
D. Zero
Explanation: Since (q+p)ⁿ = 1, multiplying by np simply gives np.
Q19. A binomial random variable can take only:
A. Non-negative values ✓ Correct Answer
B. Negative values
C. Fractional values
D. None
Explanation: Since it counts successes, a binomial variable can only
take whole, non-negative values (0, 1, 2, ...).
Q20. If X ~ b(x; 5, 0.6), then P(X = 1/2) =
A. 1.2
B. 5
C. 5.6
D. Zero ✓ Correct Answer
Explanation: A binomial variable can only take integer values, so P(X
= 1/2), a non-integer, is zero.
Q21. If X ~ b(x; 5, 0.6), then P(X = -2) =
A. 1.2
B. 0.96
C. 0.30
D. Zero ✓ Correct Answer
Explanation: A binomial variable can never be negative, so P(X = -2)
is zero.
Q22. In the binomial expansion (4/5 + 1/5)⁶, n =
A. 6 ✓ Correct Answer
B. 4/5
C. 1/5
D. 1
Explanation: The exponent in the binomial expansion directly
represents n, the number of trials — here n = 6.
Q23. The Binomial distribution has how many
parameters?
A. One
B. Two ✓ Correct Answer
C. Three
D. Four
Explanation: The binomial distribution is fully defined by two
parameters: n (number of trials) and p (probability of success).
Q24. In a Binomial Distribution, ⁷C₄(1/3)⁴(2/3)⁷⁻⁴,
then p =
A. 1/3 ✓ Correct Answer
B. 2/3
C. 7
D. ⁷C₄
Explanation: By convention, p is the probability of success, which
here corresponds to the (1/3) term raised to the number of successes (4).
Q25. The shape of the binomial distribution depends
upon the:
A. Number of trials
B. Probability of success
C. Both (a) and (b) ✓ Correct Answer
D. Values of the variable
Explanation: Both n (number of trials) and p (probability of success)
jointly determine the shape of the binomial distribution.
Q26. If p = q = 1/2, the Binomial Distribution is:
A. Asymmetrical
B. Skewed
C. Not symmetrical
D. Symmetrical ✓ Correct Answer
Explanation: When success and failure are equally likely (p=q=1/2),
the binomial distribution is perfectly symmetrical.
Q27. If p ≠ q, the Binomial Distribution is:
A. Asymmetrical
B. Skewed
C. Symmetrical
D. Both (a) and (b) ✓ Correct Answer
Explanation: Unequal probabilities of success and failure make the
distribution both asymmetrical and skewed — the same idea.
Q28. If p > 1/2, the Binomial Distribution is:
A. Positively skewed
B. Negatively skewed ✓ Correct Answer
C. Symmetrical
D. None
Explanation: When success is more likely than failure (p>1/2), the
distribution's longer tail points toward smaller values — negatively skewed.
Q29. If p < 1/2, the Binomial Distribution is:
A. Positively skewed ✓ Correct Answer
B. Negatively skewed
C. Symmetrical
D. None
Explanation: When success is less likely than failure (p<1/2), the
distribution's longer tail points toward larger values — positively skewed.
Q30. If p < q in a Binomial distribution, then:
A. Mean = median = mode
B. Mean > median >
mode ✓ Correct Answer
C. Mean < median < mode
D. None
Explanation: p<q means p<1/2, giving a positively skewed
distribution, where mean > median > mode.
Q31. If p > q in a Binomial distribution, then:
A. Mean = median = mode
B. Mean > median > mode
C. Mean < median <
mode ✓ Correct Answer
D. None
Explanation: p>q means p>1/2, giving a negatively skewed
distribution, where mean < median < mode.
Q32. The mean, median, and mode of a Binomial
distribution are equal if:
A. p = q = 1/2 ✓ Correct Answer
B. p < 1/2
C. p > 1/2
D. None
Explanation: Only when the distribution is perfectly symmetrical
(p=q=1/2) do mean, median, and mode all coincide.
Q33. Mean > Median > Mode in a Binomial
distribution if:
A. p = q = 1/2
B. p < 1/2 ✓ Correct Answer
C. p > 1/2
D. None
Explanation: A positively skewed distribution (which occurs when
p<1/2) has the ordering mean > median > mode.
Q34. Mean < Median < Mode in a Binomial
distribution if:
A. p = q = 1/2
B. p < 1/2
C. p > 1/2 ✓ Correct Answer
D. None
Explanation: A negatively skewed distribution (which occurs when
p>1/2) has the ordering mean < median < mode.
Q35. In a Binomial distribution, the mean is:
A. Greater than the
variance ✓ Correct Answer
B. Less than the variance
C. Equal to the variance
D. None
Explanation: Since mean = np and variance = npq with q<1, the
variance (npq) is always smaller than the mean (np).
Q36. The standard deviation of a Binomial
Distribution is:
A. npq
B. np
C. √(npq) ✓ Correct Answer
D. None
Explanation: Standard deviation is the square root of variance, and
variance of a binomial distribution is npq.
Q37. The mean of the Binomial distribution is:
A. np ✓ Correct Answer
B. n/p
C. p/n
D. None
Explanation: The mean of a binomial distribution is given by np
(number of trials × probability of success).
Q38. A Binomial variable with n = 6, p = 0.2 has the
standard deviation:
A. 0.2
B. 0.98 ✓ Correct Answer
C. 1.2
D. 0.33
Explanation: S.D. = √(npq) = √(6 × 0.2 × 0.8) = √0.96 ≈ 0.98.
Q39. The mean of (q + p)ⁿ is:
A. np ✓ Correct Answer
B. n/p
C. p/n
D. None
Explanation: Since (q+p)ⁿ represents the binomial expansion, its
associated mean is np.
Q40. The variance of (q + p)ⁿ is:
A. npq ✓ Correct Answer
B. nq/p
C. p/nq
D. None
Explanation: The variance associated with the binomial expansion
(q+p)ⁿ is npq.
Q41. ⁿCₓ πˣ(1-π)ⁿ⁻ˣ is called the Binomial:
A. Probability
distribution ✓ Correct Answer
B. Frequency distribution
C. Cumulative distribution
D. None
Explanation: This formula is the probability mass function of the
binomial probability distribution.
Q42. The mean of (q + p)³ is:
A. 3p ✓ Correct Answer
B. n/p
C. p/n
D. None
Explanation: For n=3, the mean of the binomial expansion is np = 3p.
Q43. The variance of (q + p)³ is:
A. 3pq ✓ Correct Answer
B. nq/p
C. p/nq
D. None
Explanation: For n=3, the variance of the binomial expansion is npq =
3pq.
Q44. In a Binomial Distribution with n = 4 and p =
1/2, P(X = 5) =
A. 1/2
B. 1/3
C. 2/3
D. Zero ✓ Correct Answer
Explanation: Since n=4, X can only range from 0 to 4 — X=5 is
impossible, so its probability is zero.
Q45. A hypergeometric experiment is one that consists
of n Bernoulli trials that are:
A. Dependent trials
B. Independent trials
C. Without replacement trials
D. Both (a) and (c) ✓ Correct Answer
Explanation: Hypergeometric trials involve sampling without
replacement, which makes successive trials dependent on each other.
Q46. In a Hypergeometric Distribution, each trial
has:
A. Two outcomes ✓ Correct Answer
B. Three outcomes
C. Four outcomes
D. None
Explanation: Like the binomial, each trial in a hypergeometric setting
has just two outcomes: success or failure.
Q47. Both Binomial and Hypergeometric distributions
are:
A. Mixed
B. Continuous
C. Discrete ✓ Correct Answer
D. None
Explanation: Since both distributions count whole-number successes,
they are discrete probability distributions.
Q48. A changing probability of success at each trial
is characteristic of:
A. Binomial Distribution
B. Uniform Distribution
C. Hypergeometric
Distribution ✓ Correct Answer
D. Selection with replacement
Explanation: Because sampling is done without replacement, the
probability of success changes after each draw in a hypergeometric setting.
Q49. Successive trials being dependent is
characteristic of:
A. Binomial Distribution
B. Selection without replacement
C. Hypergeometric Distribution
D. Both (b) and (c) ✓ Correct Answer
Explanation: Selection without replacement (the basis of the
hypergeometric distribution) makes each trial dependent on the previous ones.
Q50. Drawing two cards without replacement follows a:
A. Binomial Distribution
B. Uniform Distribution
C. Hypergeometric
Distribution ✓ Correct Answer
D. None
Explanation: Without replacement, the composition of the deck changes
between draws — the hallmark of a hypergeometric setup.
Q51. A probability of success that varies from trial
to trial is characteristic of:
A. Binomial Distribution
B. Uniform Distribution
C. Hypergeometric
Distribution ✓ Correct Answer
D. None
Explanation: Unlike the binomial (constant p), the hypergeometric
distribution has a probability of success that changes with each trial.
Q52. Successive trials being without replacement is
characteristic of:
A. Hypergeometric
Distribution ✓ Correct Answer
B. Uniform Distribution
C. Binomial Distribution
D. None
Explanation: Sampling without replacement is exactly the defining
condition of the hypergeometric distribution.
Q53. A random sample chosen without replacement from
a finite population follows a:
A. Binomial Distribution
B. Uniform Distribution
C. Hypergeometric Distribution ✓ Correct Answer
D. None
Explanation: Without-replacement sampling from a finite population is
the defining scenario for the hypergeometric distribution.
Q54. A random sample chosen by selecting all items at
once from a finite population follows a:
A. Binomial Distribution
B. Hypergeometric
Distribution ✓ Correct Answer
C. Uniform Distribution
D. None
Explanation: Selecting an entire sample at once (equivalent to without
replacement) corresponds to the hypergeometric distribution.
Q55. The Hypergeometric Distribution has how many
parameters?
A. One
B. Two
C. Three ✓ Correct Answer
D. Four
Explanation: The hypergeometric distribution is defined by three
parameters: N (population size), n (sample size), and k (number of successes in
the population).
Q56. The mean of the Hypergeometric Distribution is:
A. nk/N ✓ Correct Answer
B. N/nk
C. (nk/N)[(N-k)/N]
D. None
Explanation: The mean of a hypergeometric distribution is given by
nk/N.
Q57. The variance of the Hypergeometric Distribution
is:
A.
(nk/N)[(N-k)/N][(N-n)/(N-1)] ✓ Correct
Answer
B. N/nk
C. nk/N
D. None
Explanation: This is the standard formula for the variance of a
hypergeometric distribution, incorporating the finite population correction.
Q58. The standard deviation of the Hypergeometric
Distribution is:
A. Np
B.
√{(nk/N)[(N-k)/N][(N-n)/(N-1)]} ✓
Correct Answer
C. npq
D. None
Explanation: Standard deviation is the square root of the
hypergeometric variance formula.
Q59. If X ~ h(x; N, n, k), the mean is:
A. nk/N ✓ Correct Answer
B. np
C. npq
D. None
Explanation: For a hypergeometric random variable, the mean is given
by nk/N.
Q60. Which of the following is NOT an assumption of
the Binomial distribution?
A. All trials must be
dependent ✓ Correct Answer
B. Each trial must be classified
as a success or a failure
C. The number of successes in the
trials is counted
D. Probability of success remains
fixed
Explanation: The binomial distribution actually requires INDEPENDENT
trials — so 'all trials must be dependent' is the false statement, not an
actual assumption.
Q61. Suppose 60% of a group of rats is infected with
a disease. Let X = the number of diseased rats in a sample of size 5. The
distribution of X is:
A. Binomial with n=5 and
p=0.6 ✓ Correct Answer
B. Binomial with n=5 and p=0.4
C. Binomial with n=5 and p=0.5
D. The same as the distribution
of the number of uninfected rats
Explanation: Since each rat is independently infected with probability
0.6, and we're counting successes in 5 trials, X follows Binomial(n=5, p=0.6).
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