Probability I — 2025A Past Questions

University of Ilorin STA112: 62 questions.

Topics: Random variables, Basic distributions: Uniform, Concepts and principles of probability, Basic distributions: Poisson, Basic distributions, Basic distributions: Normal, Permutation and combination, Basic distributions: Binomial, Basic distributions: Geometric

1. In a family of 6, what is the probability of having 0 boys?
   1. 0.03125
   2. 0.01562
   3. 0.12500
   4. 0.06250
2. There are 3 Mathematics, 4 French, and 2 Physics books to be arranged on a shelf. How many ways can this be done if only the French books must be together?
   1. 720
   2. 1440
   3. 2880
   4. 5760
3. Which distribution has its values contained in an interval?
   1. Normal
   2. Binomial
   3. Uniform
   4. Poisson
4. A coin is tossed three times. What is the probability of obtaining at least two heads?
   1. 1/2
   2. 3/8
   3. 5/8
   4. 3/4
5. A coin is tossed three times. What is the probability of obtaining a head in the second attempt?
   1. 1
   2. 0
   3. 1/2
   4. 1/4
6. A journey takes a mean time of 5.5 hours with standard deviation of 0.5 hours. What is the probability that it takes between 5 and 6 hours, assuming a normal distribution?
   1. 0.6826
   2. 0.7745
   3. 0.9544
   4. 0.8413
7. Which probability distribution is often used for skewed distributions?
   1. Poisson
   2. Normal
   3. Binomial
   4. Uniform
8. A uniform distribution is distributed across an interval of [3,13]. Find the probability density function.
   1. 0.05
   2. 0.1
   3. 0.2
   4. 0.3
9. Which of the following distributions does not use a probability density function?
   1. Binomial
   2. Poisson
   3. Normal
   4. Geometric
10. What is the set of all possible outcomes of an experiment called?
    1. Sample space
    2. Event space
    3. Experiment set
    4. Trial set
11. Which of the following distributions has the property used to study skewed populations?
    1. Poisson
    2. Normal
    3. Binomial
    4. Uniform
12. A fair coin is tossed 3 times and the sequence of outcomes is observed. What is the probability that the first and third tosses are heads?
    1. 0.500
    2. 0.250
    3. 0.375
    4. 0.875
13. If X has a uniform distribution over the interval [2,13], find P(x < 7).
    1. 5/11
    2. 4/11
    3. 6/11
    4. 7/11
14. If X has a uniform distribution over the interval [2,13], find P(x < 6).
    1. 2/11
    2. 3/11
    3. 4/11
    4. 5/11
15. If X follows a uniform distribution over the interval [2,10], find the mean of X.
    1. 5
    2. 6
    3. 7
    4. 8
16. If X has a uniform distribution over the interval [3,12], find P(x < 9).
    1. 5/9
    2. 6/9
    3. 7/9
    4. 8/9
17. Which of the following distributions has standard deviation √λ?
    1. Poisson
    2. Binomial
    3. Geometric
    4. Uniform
18. If Var(x + 5t) equals what? (Assuming t is a constant)
    1. Var(x)
    2. Var(x) + t
    3. Var(x) + 25
    4. Var(x) + 5t
19. If the probability that a UTME student gets admitted into Medicine is 0.6, what is the expected number of students admitted from 120 applicants?
    1. 69
    2. 72
    3. 75
    4. 78
20. If X is uniformly distributed over [3,12], find P(x > 6).
    1. 6/9
    2. 5/9
    3. 4/9
    4. 3/9
21. In a family of 3 children, what is the probability of having 2 boys?
    1. 1/8
    2. 3/8
    3. 1/2
    4. 1/4
22. Two dice were thrown once simultaneously, what is the probability of obtaining a sum of 7?
    1. 1/6
    2. 1/12
    3. 1/9
    4. 1/4
23. An average of 2 in every 50,000 insects per year have death records. In a colony of 100,000, what is the probability that none of them would die? (Assuming Poisson distribution)
    1. 0.1353
    2. 0.2707
    3. 0.3679
    4. 0.5000
24. An average of 2 in every 50,000 insects per year have death records. In a colony of 100,000, what is the value for lambda?
    1. 2
    2. 1
    3. 4
    4. 5
25. A coin was thrown thrice. What is the probability that the second toss was head?
    1. 1/2
    2. 1/4
    3. 1
    4. 0
26. Which distribution is used for a skewed population?
    1. Poisson
    2. Bernoulli
    3. Binomial
    4. Geometric
27. If X and Y are dependent random variables, the f(y|x) is?
    1. Joint probability
    2. Conditional probability
    3. Marginal probability
    4. Independent probability
28. Given a distribution f(x) = x²/91, for x = 1, 2, 3, 4, 5, 6. Find P(x = 5).
    1. 25/91
    2. 5/91
    3. 1/91
    4. 10/91
29. A bag contains 2 white and 3 red balls. How many ways can 3 balls be chosen if at least one ball must be white?
    1. 10
    2. 8
    3. 9
    4. 7
30. X follows a uniform distribution over the interval (0,11). Find Pr(3 < x < 7).
    1. 4/11
    2. 1/2
    3. 7/11
    4. 3/11
31. What distribution is used for skewed population?
    1. Poisson
    2. Binomial
    3. Normal
    4. Uniform
32. A fair coin is tossed three times. Find the probability of obtaining at most two tails.
    1. 7/8
    2. 3/4
    3. 1/4
    4. 1/2
33. A fair coin is tossed three times. Find the probability of getting head in the second toss.
    1. 1/2
    2. 1/3
    3. 3/4
    4. 1/4
34. What distribution is given by λ^x * e^(-λ) / x! ?
    1. Poisson
    2. Binomial
    3. Normal
    4. Geometric
35. Normally distributed over the mean and variance of the weight are 45kg and 122kg. What is the probability that the weight is exactly 45kg?
    1. 0.01
    2. 0
    3. 0.5
    4. 1
36. Which of the following is not part of a discrete distribution?
    1. Geometric
    2. Binomial
    3. Normal
    4. Poisson
37. In a bag contains 2 white balls and 3 red balls. How many ways can 3 balls be chosen if at least one must be red?
    1. 9
    2. 10
    3. 12
    4. 15
38. In a family of 6, what is the probability of having 2 boys? (Assuming equal probability)
    1. 0.3438
    2. 0.1094
    3. 0.2434
    4. 0.2344
39. Which of these is not a random variable?
    1. Weight
    2. Height
    3. Name
    4. Truth
40. If X ~ Bi(6, 0.3). Find the probability that X > 2.
    1. 0.6172
    2. 0.8504
    3. 0.4221
    4. 0.3779
41. Given a distribution f(x) = x² / 91, x = 1, 2, 3, 4, 5, 6. Find Pr(x = 1).
    1. 1/91
    2. 1/13
    3. 1/7
    4. 1/6
42. For f(x) = x² / 91, x = 1 to 6. Find Pr(x ≥ 5).
    1. 61/91
    2. 61/100
    3. 61/121
    4. 61/81
43. What is the formula for geometric variance?
    1. q/p²
    2. pq
    3. p/q²
    4. 1/p
44. How many different numbers of 6 digits can be formed using 4,4,6,6,6,6?
    1. 90
    2. 150
    3. 60
    4. 180
45. Probability of having a sum of 7 when two dice are thrown together is:
    1. 6/36
    2. 8/36
    3. 7/36
    4. 5/36
46. Under normal distribution, if Unilorin students' average weight is 45 and variance is 121, what is the probability the student weighs exactly 45?
    1. 0
    2. 0.5
    3. 1
    4. 0.25
47. Under normal distribution, if Unilorin students' average weight is 45 and variance is 121, what is the probability the student weighs at least 40?
    1. 0.8413
    2. 0.9772
    3. 0.5
    4. 0.6915
48. Uniform distribution: Find P(x > 5) if X ~ U(3,13).
    1. 4/5
    2. 1/5
    3. 8/13
    4. 2/3
49. Which distribution has standard deviation of (b - a)/√12?
    1. Normal distribution
    2. Uniform distribution
    3. Geometric distribution
    4. Probability distribution
50. Find the variance of a uniform distribution over (2, 10).
    1. 16/3
    2. 64/12
    3. 8
    4. 18
51. In a general hospital, 10 nurses are to be assigned: 5 to wards, 3 to pharmacy, and 2 to the laboratory. In how many ways can these 10 nurses be assigned?
    1. 2520
    2. 3580
    3. 2100
    4. 1200
52. Given that the expected number of phone calls is 4 per minute, what is the probability that there will be exactly 2 calls in 30 seconds? (Assume Poisson distribution)
    1. 0.1353
    2. 0.3679
    3. 0.019
    4. 0.2707
53. Calculate the mean and variance of a uniform distribution over (2,13).
    1. Mean = 7.5, Variance = 10.08
    2. Mean = 7, Variance = 8.33
    3. Mean = 8, Variance = 11
    4. Mean = 6, Variance = 9.17
54. Using a Poisson distribution, calculate the probability that x = 3 given that λ (mean) is 2.11.
    1. 0.186
    2. 0.214
    3. 0.251
    4. 0.164
55. What is the definition of a random variable?
    1. A value assigned to an event based on frequency
    2. A numerical outcome of a random experiment
    3. A number that always changes
    4. A fixed unknown in an equation
56. The formula for normal distribution is:
    1. (1 / √2π) * e^(-x2/2)
    2. e^(λx)
    3. 1/n Σ(x - μ)2
    4. (1 / σ√2π) * e^(-(x - μ)2 / 2σ2)
57. Given two dependent variables f(x|y), which is correct?
    1. f(x) + f(y)
    2. f(x) - f(y)
    3. f(x|y) ≠ f(x)
    4. f(x|y) = f(y)
58. Given a Bernoulli distribution where P = 16/30 and Q = 14/30, what is the expected value (mean)?
    1. 0.533
    2. 0.467
    3. 1
    4. 0.5
59. A group of rats are subjected to a response test which can be either positive or negative. If the stimulus is applied to three rats in succession, what is the probability that both the first and the last respond positively? (Assume responses are independent and probability of positive is 1/2)
    1. 1/4
    2. 1/2
    3. 1/8
    4. 1/6
60. If a lady divorces 9 times, what is the probability that she will remarry within 3 years?
    1. 1
    2. Cannot be determined
    3. 0.5
    4. 0.75
61. In a certain town, 4% of couples are divorced. If a chancellor interviews couples one by one at random, how many couples is he expected to interview before he finds the first divorced couple?
    1. 25
    2. 4
    3. 20
    4. 10
62. A person rolls a fair six-sided die repeatedly. What is the probability that they will roll a 6 for the first time between the 4th and 6th trial?
    1. 0.096
    2. 0.312
    3. 0.421
    4. 0.5

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