1. What is the probability of getting a head when a fair coin is tossed?
A) 0
B) 1/2
C) 1
D) 2/3
Answer: B) 1/2
Explanation: Two outcomes → {Head, Tail}. One favorable → Probability = 1/2.
2. Probability of an impossible event is:
A) 1
B) 0
C) 1/2
D) Undefined
Answer: B) 0
Explanation: Impossible event has no favorable outcomes.
3. Probability of a sure event is:
A) 0
B) 1/2
C) 1
D) Cannot be determined
Answer: C) 1
Explanation: A certain event always occurs → probability = 1.
4. If P(E) = 0.7, then probability of not E is:
A) 0.3
B) 0.5
C) 0.7
D) 1.7
Answer: A) 0.3
Explanation: P(not E) = 1 – P(E) = 1 – 0.7 = 0.3.
5. Which of the following is not a valid probability value?
A) 0
B) 1/2
C) 3/4
D) 1.2
Answer: D) 1.2
Explanation: Probability ∈ [0, 1].
6. A dice is thrown. Probability of getting a number greater than 4?
A) 1/6
B) 2/6
C) 1/3
D) 1/2
Answer: C) 1/3
Explanation: Outcomes > 4 → {5, 6} → 2/6 = 1/3.
7. A card is drawn from a pack of 52. Probability of drawing a red card?
A) 1/4
B) 1/2
C) 1/3
D) 2/3
Answer: B) 1/2
Explanation: Red cards = 26/52 = 1/2.
8. Probability of getting an odd number when a dice is thrown?
A) 1/6
B) 1/2
C) 2/3
D) 5/6
Answer: B) 1/2
Explanation: Odd numbers {1,3,5} → 3/6 = 1/2.
9. If a coin is tossed twice, total possible outcomes = ?
A) 2
B) 3
C) 4
D) 6
Answer: C) 4
Explanation: Sample space {HH, HT, TH, TT}.
10. Probability of getting at least one head when two coins are tossed?
A) 1/4
B) 1/2
C) 3/4
D) 1
Answer: C) 3/4
Explanation: Outcomes = 4. Favourable = {HH, HT, TH} = 3. → 3/4.
11. A number is chosen at random from 1 to 10. Probability it is prime?
A) 2/5
B) 3/5
C) 4/5
D) 1/2
Answer: A) 2/5
Explanation: Primes ≤ 10: {2,3,5,7} → 4/10 = 2/5.
12. If a card is drawn, probability it is a King?
A) 1/52
B) 1/26
C) 1/13
D) 1/12
Answer: C) 1/13
Explanation: Total Kings = 4. Probability = 4/52 = 1/13.
13. What is the probability of getting number 7 on a dice?
A) 0
B) 1/6
C) 1/12
D) 1/7
Answer: A) 0
Explanation: Dice outcomes 1–6. 7 is impossible → probability 0.
14. In tossing 3 coins, probability of getting all heads?
A) 1/8
B) 1/4
C) 1/2
D) 3/8
Answer: A) 1/8
Explanation: Outcomes = 8. Only 1 favorable (HHH). → 1/8.
15. If two dice are thrown, total sample space size = ?
A) 6
B) 12
C) 18
D) 36
Answer: D) 36
Explanation: Each die 6 outcomes. 6×6 = 36.
16. Probability of getting a sum of 7 on two dice?
A) 1/6
B) 1/12
C) 1/18
D) 1/36
Answer: A) 1/6
Explanation: Pairs giving 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) → 6/36 = 1/6.
17. Probability of drawing a black card?
A) 1/2
B) 1/4
C) 2/3
D) 1/3
Answer: A) 1/2
Explanation: Black cards = 26. Probability = 26/52 = 1/2.
18. A die is rolled. Probability of a multiple of 3?
A) 1/6
B) 1/3
C) 1/2
D) 2/3
Answer: B) 1/3
Explanation: Multiples of 3 = {3,6} → 2/6 = 1/3.
19. If P(E) = 0.35, then P(not E) = ?
A) 0.45
B) 0.55
C) 0.65
D) 0.75
Answer: C) 0.65
Explanation: 1 – 0.35 = 0.65.
20. Probability of selecting a vowel from English alphabets (A–Z)?
A) 5/26
B) 21/26
C) 1/5
D) 5/21
Answer: A) 5/26
Explanation: Vowels = {A,E,I,O,U} → 5/26.
21. A card is drawn. Probability of drawing an ace?
A) 1/52
B) 1/13
C) 1/26
D) 4/13
Answer: B) 1/13
Explanation: 4 aces in 52 cards → 4/52 = 1/13.
22. In a lottery of 1000 tickets, one ticket wins. If you buy 1 ticket, probability of winning = ?
A) 1/1000
B) 1/100
C) 1/10
D) 0
Answer: A) 1/1000
Explanation: Favourable = 1, total = 1000 → 1/1000.
23. A bag has 3 red and 2 black balls. Probability of drawing a red ball?
A) 2/5
B) 3/5
C) 1/2
D) 1/3
Answer: B) 3/5
Explanation: Total = 5, Red = 3 → 3/5.
24. A die is thrown. Probability of getting even number?
A) 1/2
B) 1/3
C) 2/3
D) 5/6
Answer: A) 1/2
Explanation: Even {2,4,6} → 3/6 = 1/2.
25. Which is correct for any event E?
A) 0 ≤ P(E) ≤ 1
B) –1 ≤ P(E) ≤ 1
C) P(E) ≥ 0 always
D) P(E) ≥ 1 always
Answer: A) 0 ≤ P(E) ≤ 1
Explanation: Probability always lies between 0 and 1.
26. Two dice are thrown. Probability of getting doublets (same number on both dice)?
A) 1/6
B) 1/12
C) 1/18
D) 1/36
Answer: A) 1/6
Explanation: Doublets = (1,1),(2,2)…(6,6) → 6 outcomes. Probability = 6/36 = 1/6.
27. Probability of getting a sum of 11 on two dice?
A) 1/18
B) 1/12
C) 1/6
D) 1/36
Answer: A) 1/18
Explanation: Pairs = (5,6),(6,5) → 2 outcomes. Probability = 2/36 = 1/18.
28. A card is drawn. Probability it is neither king nor queen?
A) 12/13
B) 25/26
C) 23/26
D) 11/13
Answer: A) 12/13
Explanation: Kings = 4, Queens = 4 → 8 total. Favorable = 52–8=44. Probability = 44/52=11/13.
29. A coin is tossed 3 times. Probability of getting exactly 2 heads?
A) 1/8
B) 3/8
C) 3/4
D) 1/2
Answer: B) 3/8
Explanation: Outcomes = 8. Favourable = {HHT, HTH, THH} = 3. Probability = 3/8.
30. A card is drawn from 52. Probability it is a spade or a king?
A) 4/13
B) 17/52
C) 16/52
D) 13/52
Answer: B) 17/52
Explanation: Spades = 13, Kings = 4, but overlap (king of spades counted twice). Total favorable = 13+4–1=16. Probability = 16/52 = 4/13.
31. Probability of getting at least one 6 in two dice throws?
A) 5/36
B) 11/36
C) 25/36
D) 1/6
Answer: C) 25/36
Explanation: P(no 6) = (5/6)×(5/6)=25/36. So P(at least one 6)=1–25/36=11/36.
(⚠ Correction: Actually → 11/36.)
✔ Correct Answer: B) 11/36
32. A coin is tossed 4 times. Probability of getting all tails?
A) 1/8
B) 1/16
C) 1/32
D) 1/4
Answer: B) 1/16
Explanation: Total = 16 outcomes. Only TTTT favorable. Probability = 1/16.
33. A card is drawn. Probability it is a red face card?
A) 3/13
B) 3/26
C) 6/52
D) 12/52
Answer: B) 3/26
Explanation: Face cards = 12 (J,Q,K). Red face cards = 6 (♥♦). Probability = 6/52=3/26.
34. Two coins are tossed. Probability of getting at least one tail?
A) 1/2
B) 3/4
C) 1/4
D) 2/3
Answer: B) 3/4
Explanation: Total outcomes = 4. Favourable = {HT,TH,TT}=3. Probability=3/4.
35. Probability of getting a sum of 9 on two dice?
A) 1/9
B) 1/8
C) 1/12
D) 1/18
Answer: C) 1/12
Explanation: Outcomes = (3,6),(4,5),(5,4),(6,3) → 4. Probability=4/36=1/9.
✔ Correct Answer: A) 1/9
36. A card is drawn. Probability it is not a heart?
A) 1/2
B) 3/4
C) 1/3
D) 2/3
Answer: B) 3/4
Explanation: Hearts=13, not hearts=39. Probability=39/52=3/4.
37. Probability of getting exactly one head in 3 coin tosses?
A) 1/8
B) 2/8
C) 3/8
D) 4/8
Answer: C) 3/8
Explanation: Favourable = {HTT, THT, TTH} → 3. Probability=3/8.
38. If two dice are thrown, probability that both numbers are even?
A) 1/4
B) 1/9
C) 1/3
D) 1/6
Answer: A) 1/4
Explanation: Even on each die=3/6=1/2. Together=(1/2)×(1/2)=1/4.
39. Probability of drawing a black queen from 52 cards?
A) 1/26
B) 1/52
C) 1/13
D) 1/12
Answer: A) 1/26
Explanation: Black queens=2. Probability=2/52=1/26.
40. Two dice are thrown. Probability of getting an even sum?
A) 1/4
B) 1/2
C) 2/3
D) 3/4
Answer: B) 1/2
Explanation: Even sum occurs if both even or both odd. Half the outcomes are even → 18/36=1/2.
41. A coin is tossed. Probability of not getting a tail?
A) 0
B) 1/2
C) 1
D) 1/3
Answer: B) 1/2
Explanation: Not tail= head → probability 1/2.
42. Probability of drawing a king of spades?
A) 1/52
B) 1/26
C) 1/13
D) 1/12
Answer: A) 1/52
Explanation: Only one king of spades. Probability=1/52.
43. If 2 dice are thrown, probability of both numbers being prime?
A) 1/9
B) 1/6
C) 1/4
D) 1/3
Answer: A) 1/9
Explanation: Prime numbers on dice= {2,3,5} =3. Probability=3/6=1/2 each. Combined=1/4.
Correction: Actually → (3/6)×(3/6)=9/36=1/4.
✔ Correct Answer: C) 1/4
44. Probability of getting 2 or 3 on a dice?
A) 1/6
B) 2/6
C) 1/3
D) 1/2
Answer: B) 2/6 = 1/3
Explanation: Favourable={2,3}. Probability=2/6=1/3.
45. A card is drawn. Probability that it is neither red nor a king?
A) 3/4
B) 11/13
C) 12/13
D) 25/26
Answer: B) 11/13
Explanation: Red cards=26, kings=4, overlap=2 (red kings). Total=28. Favorable=24. Probability=24/52=6/13.
✔ Correct Answer: 6/13 (not in given options)
46. Two dice thrown. Probability of getting sum ≤ 4?
A) 1/9
B) 1/6
C) 1/12
D) 1/18
Answer: B) 1/6
Explanation: Favourable sums=2,3,4. Outcomes=(1,1),(1,2),(2,1),(1,3),(2,2),(3,1) →6. Probability=6/36=1/6.
47. Probability of drawing a non-face card?
A) 10/13
B) 11/13
C) 3/4
D) 2/3
Answer: A) 10/13
Explanation: Face cards=12. Non-face=40. Probability=40/52=10/13.
48. A card is drawn. Probability it is a diamond or club?
A) 1/2
B) 1/4
C) 1/3
D) 2/3
Answer: A) 1/2
Explanation: Diamonds=13, clubs=13 → 26. Probability=26/52=1/2.
49. Probability of getting a sum of 5 on two dice?
A) 1/9
B) 1/12
C) 1/18
D) 1/36
Answer: B) 1/12
Explanation: Outcomes: (1,4),(2,3),(3,2),(4,1)=4. Probability=4/36=1/9.
✔ Correct Answer: A) 1/9
50. If two coins are tossed, probability of getting one head and one tail?
A) 1/4
B) 1/2
C) 3/4
D) 2/3
Answer: B) 1/2
Explanation: Outcomes= {HH,HT,TH,TT}. Favorable={HT,TH}=2. Probability=2/4=1/2.
51. If P(A) = 0.5, P(B) = 0.4 and A & B are independent, find P(A ∩ B).
A) 0.1
B) 0.2
C) 0.25
D) 0.9
Answer: B) 0.2
Explanation: For independent events → P(A ∩ B) = P(A) × P(B) = 0.5 × 0.4 = 0.2.
52. If P(A) = 0.7, P(B) = 0.6, P(A ∩ B) = 0.5, then P(A ∪ B) = ?
A) 0.8
B) 0.9
C) 1.2
D) 0.7
Answer: A) 0.8
Explanation: P(A ∪ B) = P(A)+P(B)–P(A ∩ B) = 0.7+0.6–0.5=0.8.
53. Two independent events A and B have probabilities 0.6 and 0.5. Probability that both occur?
A) 0.1
B) 0.2
C) 0.3
D) 0.25
Answer: C) 0.3
Explanation: P(A ∩ B) = 0.6×0.5 = 0.3.
54. If P(A) = 0.4, P(B) = 0.5, and A, B independent, probability that neither occurs?
A) 0.1
B) 0.2
C) 0.3
D) 0.6
Answer: D) 0.6
Explanation: P(A ∪ B) = 0.4+0.5–0.2=0.7. So P(none) = 1–0.7=0.3.
✔ Correct Answer: C) 0.3
55. If P(A) = 0.3, P(B) = 0.2, and A,B mutually exclusive, P(A ∪ B) = ?
A) 0.1
B) 0.3
C) 0.5
D) 0.6
Answer: C) 0.5
Explanation: Mutually exclusive → P(A ∩ B)=0. So P(A ∪ B)=0.3+0.2=0.5.
56. If P(A) = 0.4, P(B) = 0.5, P(A ∩ B) = 0.2, then P(A|B) = ?
A) 0.2
B) 0.4
C) 0.6
D) 0.7
Answer: C) 0.6
Explanation: P(A|B) = P(A ∩ B)/P(B) = 0.2/0.5=0.4.
✔ Correct Answer: B) 0.4
57. If two dice are thrown, find P(sum is even | first die shows 3).
A) 1/2
B) 1/3
C) 2/3
D) 5/6
Answer: A) 1/2
Explanation: If first die=3 (odd), second die must be odd for sum even. Half the outcomes → 3/6=1/2.
58. A bag has 3 red, 2 black. One ball drawn. Probability red given it is not black?
A) 1
B) 3/5
C) 1/2
D) 2/3
Answer: A) 1
Explanation: If not black → must be red. Probability = 1.
59. A coin tossed twice. Find P(first toss is head | exactly one head occurs).
A) 1/2
B) 1/3
C) 2/3
D) 1/4
Answer: A) 1/2
Explanation: Exactly one head outcomes {HT,TH}. Given condition → 2 outcomes. Favourable {HT}. Probability=1/2.
60. A bag has 2 white, 3 red, 5 black. One ball drawn. P(red|not black)?
A) 1/5
B) 1/2
C) 3/5
D) 3/7
Answer: D) 3/7
Explanation: Not black → {2W,3R}=5. Red among them=3. Probability=3/5.
✔ Correct Answer: C) 3/5
61. If P(A) = 0.5, P(B) = 0.3, P(A ∩ B) = 0.2, find P(A|B).
A) 2/3
B) 1/2
C) 1/3
D) 1/4
Answer: A) 2/3
Explanation: P(A|B)=P(A ∩ B)/P(B)=0.2/0.3=2/3.
62. Two cards drawn without replacement from 52. Find P(both aces).
A) 1/169
B) 1/221
C) 1/1326
D) 1/26
Answer: B) 1/221
Explanation: P= (4/52)×(3/51) = 12/2652 = 1/221.
63. From 1–20, one number chosen. Find P(multiple of 4 | even).
A) 1/2
B) 1/3
C) 2/5
D) 1/4
Answer: B) 1/3
Explanation: Even numbers=10. Multiples of 4 among them= {4,8,12,16,20}=5. Probability=5/10=1/2.
✔ Correct Answer: A) 1/2
64. If P(A) = 0.6, P(B) = 0.5, and P(A ∩ B)=0.3, are A and B independent?
A) Yes
B) No
C) Sometimes
D) Can’t say
Answer: A) Yes
Explanation: Independence check: P(A)×P(B)=0.6×0.5=0.3 = P(A ∩ B). So independent.
65. A bag has 3 white, 2 red. Two balls drawn without replacement. Find P(both white).
A) 1/5
B) 1/10
C) 1/3
D) 3/10
Answer: D) 3/10
Explanation: P= (3/5)×(2/4)=6/20=3/10.
66. If a card is drawn, probability that it is king given that it is a face card?
A) 1/12
B) 1/3
C) 1/4
D) 1/13
Answer: B) 1/3
Explanation: Face cards=12 (J,Q,K). Kings=4. Probability=4/12=1/3.
67. Two dice thrown. Probability that sum is 9 given first die is 4?
A) 1/6
B) 1/3
C) 1/2
D) 1/12
Answer: A) 1/6
Explanation: If first=4, second must=5. One favourable out of 6 → 1/6.
68. A bag has 2 red, 3 green, 5 blue. One ball drawn. P(green | not blue)?
A) 1/2
B) 1/3
C) 3/5
D) 3/8
Answer: B) 1/3
Explanation: Not blue = 5 balls (2R+3G). Green among them=3. Probability=3/5.
✔ Correct Answer: C) 3/5
69. A card drawn. Find P(king or queen | face card).
A) 1/2
B) 2/3
C) 1/3
D) 1/4
Answer: B) 2/3
Explanation: Face cards=12. Kings+Queens=8. Probability=8/12=2/3.
70. A coin is tossed thrice. Find P(exactly 2 heads | at least 1 head).
A) 3/8
B) 3/7
C) 1/3
D) 1/2
Answer: B) 3/7
Explanation: Total outcomes=8. At least one head=7 outcomes. Exactly 2 heads=3 outcomes. Probability=3/7.
71. A die is rolled. Find P(even number | prime number).
A) 1/2
B) 1/3
C) 1/4
D) 2/3
Answer: C) 1/4
Explanation: Primes on die={2,3,5}. Even among them={2}. Probability=1/3.
✔ Correct Answer: B) 1/3
72. From 1–10, number chosen. P(odd | not prime)?
A) 1/2
B) 1/3
C) 1/4
D) 2/5
Answer: D) 2/5
Explanation: Non-primes= {1,4,6,8,9,10} = 6. Odd among them= {1,9}=2. Probability=2/6=1/3.
✔ Correct Answer: B) 1/3
73. A box has 2 defective, 8 good bulbs. If 1 picked, find P(defective | not good)?
A) 0
B) 1
C) 1/5
D) 2/5
Answer: B) 1
Explanation: If not good → must be defective. Probability=1.
74. If a coin is tossed thrice, find P(first toss head | total heads = 2).
A) 1/2
B) 1/3
C) 2/3
D) 1/4
Answer: C) 2/3
Explanation: Exactly 2 heads outcomes= {HHT,HTH,THH}. First head in 2 of them. Probability=2/3.
75. (Bayes’ theorem) A bag has 2 red, 3 green. Another has 4 red, 5 green. One bag chosen at random, 1 ball drawn and it is red. Probability it came from first bag?
A) 2/7
B) 3/7
C) 4/7
D) 5/7
Answer: B) 3/7
Explanation:
- P(Bag1)=1/2, P(R|Bag1)=2/5.
- P(Bag2)=1/2, P(R|Bag2)=4/9.
- P(R)= (1/2×2/5)+(1/2×4/9) = 23/90.
- P(Bag1|R)= [(1/2×2/5)] / (23/90) = (1/5)/(23/90)=18/90=2/10=0.2.
Correction → simplify carefully: (1/5)/(23/90) = (18/115)=0.156.
76. Five fair coins are tossed. Probability of getting exactly 3 heads is
A) 5/16
B) 5/32
C) 10/32
D) 10/16
Answer: C) 10/32 (which simplifies to 5/16)
Explanation: Number of ways = C(5,3)=10. Total outcomes = 2⁵=32. Probability = 10/32 = 5/16.
77. Two dice are rolled. Probability that the product is divisible by 3 is
A) 1/3
B) 5/9
C) 2/3
D) 8/9
Answer: B) 5/9
Explanation: Product divisible by 3 unless both dice are ≠ multiples of 3. Numbers not divisible by 3: {1,2,4,5} (4 choices). Probability both not divisible by 3 = (4/6)×(4/6)= (2/3)×(2/3)=4/9. So product divisible by 3 = 1 − 4/9 = 5/9.
78. From 10 balls (4 red, 6 blue) pick 3 without replacement. Probability all three are blue =
A) C(6,3)/C(10,3)
B) (6/10)³
C) (4/10)(3/9)(2/8)
D) 1/10
Answer: A) C(6,3)/C(10,3) (= 20/120 = 1/6)
Explanation: Hypergeometric: choose 3 blue out of 6 divided by choose any 3 out of 10. Numeric: C(6,3)=20, C(10,3)=120 → 20/120 = 1/6.
79. In a classroom 60% like tea, 50% like coffee, and 30% like both. Probability a randomly chosen student likes neither =
A) 0.20
B) 0.30
C) 0.40
D) 0.60
Answer: A) 0.20
Explanation: P(tea ∪ coffee) = 0.6 + 0.5 − 0.3 = 0.8. So neither = 1 − 0.8 = 0.2.
80. A fair die is rolled 3 times. Probability that the sequence contains exactly one 6 is
A) 3 × (1/6) × (5/6)²
B) (1/6)³
C) C(3,1)/(6³)
D) (5/6)³
Answer: A) 3 × (1/6) × (5/6)²
Explanation: Choose which one of 3 rolls is 6: C(3,1)=3. Probability = 3*(1/6)*(5/6)².
81. A box contains 3 white and 2 black balls. Two balls are drawn with replacement. Probability that both are black =
A) (2/5)²
B) (2/5)(1/4)
C) C(2,2)/C(5,2)
D) 2/5
Answer: A) (2/5)² = 4/25
Explanation: With replacement each draw independent: (2/5)*(2/5)=4/25.
82. A factory produces items; each item independently is defective with probability 0.02. For a sample of 50 items, approximate probability that no item is defective (use (1 − p)^n) =
A) (0.98)^50
B) 50×0.02
C) e^(−1)
D) 1 − (0.02)⁵⁰
Answer: A) (0.98)^50 ≈ 0.363
Explanation: P(no defect) = (1 − 0.02)^50 = 0.98^50 ≈ e^{50 ln0.98} ≈ e^{-1.0101} ≈ 0.364.
83. From a standard deck, five cards are drawn. Probability of exactly one ace =
A) C(4,1)C(48,4)/C(52,5)
B) (4/52)(48/51)…
C) 4/52
D) C(4,1)/C(52,5)
Answer: A) C(4,1)C(48,4)/C(52,5)
Explanation: Choose 1 ace from 4 and 4 non-aces from 48; divide by total 5-card combinations.
84. A fair coin is tossed until first head. Expected number of tosses =
A) 1
B) 2
C) 1/p where p=1/2 → 2
D) 3
Answer: C) 2
Explanation: Geometric distribution with success prob p=1/2 → mean = 1/p = 2.
85. Two independent Poisson processes with rates 2 and 3 per hour are combined. Probability that in one hour total events = 4 is
A) e^{−5} 5⁴/4!
B) e^{−2} 2⁴/4! + e^{−3} 3⁴/4!
C) (e^{−2}2⁴/4!)(e^{−3}3⁴/4!)
D) impossible to compute
Answer: A) e^{−5} 5⁴/4!
Explanation: Sum is Poisson(2+3=5). So P(X=4)=e^{-5}5^4/4!.
86. From 8 people, two are chosen at random for posts of President and Secretary (different posts). Probability a particular person Alice gets at least one post =
A) 2/8
B) 1/4
C) 1/4? (same)
D) 1/5
Answer: B) 1/4
Explanation: Total ordered pairs = 8×7=56. Number of ordered pairs where Alice has at least one post = (Alice president & anyone secretary) 7 + (Alice secretary & anyone president) 7 = 14. Probability = 14/56 = 1/4.
87. A random permutation of 5 distinct letters. Probability that none is in its original position (derangement D₅) =
A) D₅/5! where D₅ = 44 → 44/120 = 11/30
B) 1/5!
C) 1/2
D) 24/120
Answer: A) 11/30
Explanation: Number of derangements for n=5 is 44. Probability = 44/120 = 11/30.
88. A box contains 6 good and 4 defective bulbs. Two bulbs are drawn without replacement. Probability exactly one defective =
A) 2 × (4/10)(6/9)
B) (4/10)(3/9)
C) C(4,1)C(6,1)/C(10,2)
D) Both A and C equal
Answer: D) Both A and C equal
Explanation: Calculate: 2*(4/10)(6/9)=2(24/90)=48/90=8/15. Hypergeometric C(4,1)C(6,1)/C(10,2) = 4*6/45=24/45=8/15.
89. Two boxes: Box A has 2 red/3 blue, Box B has 1 red/1 blue. Choose a box uniformly at random and pick a ball; observed ball is blue. Probability it came from Box B =
A) (1/2 × 1/2) / P(blue)
B) (1/2×1/2) / [(1/2×3/5)+(1/2×1/2)]
C) 1/2
D) 3/10
Answer: B) (1/2×1/2) / [(1/2×3/5)+(1/2×1/2)] = (1/4)/((3/10)+(1/4)) = (1/4)/(3/10+1/4) = (1/4)/( (6/20+5/20)=11/20 ) = (1/4)/(11/20)= (1/4)*(20/11)=5/11 ≈ 0.4545
Explanation: Bayes: P(B|blue)=P(B)P(blue|B)/P(blue). Numeric: P(blue)=0.5(3/5)+0.5*(1/2)=3/10+1/4=11/20. So answer 5/11.
90. A fair six-sided die is rolled repeatedly until two consecutive sixes appear. The expected number of rolls (approx) is
A) 36
B) 42
C) 36? (exact value 42)
D) 49
Answer: C) 42
Explanation: Expected waiting time for two consecutive successes with p=1/6 is (1 + q)/(p²) where q=1−p? More direct known result for waiting time for two consecutive successes: (1 + p)/p² − 1/p? Simpler: expected waiting time for two consecutive sixes equals (36+6)/? Known result: for two consecutive heads with p=1/2 expected = 6. For p=1/6 it’s (1+p)/p² = (1+1/6)/(1/36)= (7/6)*36 = 42. So 42.
91. A biased coin has P(head)=0.7. Probability of exactly 2 heads in 3 tosses =
A) C(3,2)(0.7)²(0.3)
B) (0.7)²
C) 3(0.7)(0.3)²
D) (0.3)³
Answer: A) 3 × 0.7² × 0.3 = 3 × 0.49 × 0.3 = 0.441
Explanation: Binomial formula: C(3,2)p²(1−p).
92. If X ~ Poisson(λ=2), P(X ≥ 1) =
A) 1 − e^{−2}
B) e^{−2}
C) 1 − 2e^{−2}
D) e^{−2}(1 + 2)
Answer: A) 1 − e^{−2}
Explanation: P(X=0)=e^{−2}. So P(X≥1)=1−P(0)=1−e^{−2}.
93. Random variable X has P(X=0)=0.2, P(X=1)=0.5, P(X=2)=0.3. E[X] =
A) 1.1
B) 1.0
C) 0.8
D) 1.5
Answer: A) 1.1
Explanation: E[X] = 0·0.2 + 1·0.5 + 2·0.3 = 0 + 0.5 + 0.6 = 1.1.
94. Two fair dice are rolled. Given that the sum is 8, probability that one die shows 2 is
A) 1/5
B) 1/4
C) 1/3
D) 2/5
Answer: B) 1/4
Explanation: Outcomes summing to 8: (2,6),(3,5),(4,4),(5,3),(6,2) → 5 equally likely outcomes. Ones with a 2 are (2,6),(6,2) = 2. Probability = 2/5. Wait check: that’s 2/5. So correct is D) 2/5. (Oops — corrected).
Final: D) 2/5.
95. From integers 1 to 50 pick one at random. Probability it is divisible by both 2 and 5 = divisible by 10 =
A) 4/25
B) 1/5
C) 5/50
D) 1/10
Answer: D) 1/5 (since multiples of 10 in 1..50 are 5 numbers: 10,20,30,40,50 → 5/50 = 1/10).
Explanation: Wait compute carefully: 5/50 = 1/10. So correct is D) 1/10. (Ignore the first mistaken choice.)
96. A jar contains 3 red, 4 green, 5 blue marbles. One marble drawn at random. Probability it is green or blue =
A) 9/12
B) 3/12
C) 5/12
D) 7/12
Answer: A) 9/12 = 3/4
Explanation: Green+Blue = 4+5 = 9 out of total 12 → 3/4.
97. Suppose X ~ Binomial(n=4, p=0.25). P(X=0) =
A) (0.75)^4
B) (0.25)^4
C) 4(0.25)(0.75)^3
D) 1 − (0.75)^4
Answer: A) (0.75)^4 = 0.31640625
Explanation: Probability of zero successes = (1−p)^n.
98. A password is a random permutation of the letters A,B,C,D (all distinct). Probability the password starts with A or ends with D =
A) 1/2
B) 2/4
C) 3/4
D) 7/24
Answer: A) 1/2
Explanation: Total permutations = 4! = 24. Count with starts with A: 3! = 6. Count with ends with D: 3! = 6. Overlap (starts with A and ends with D): 2! = 2. By inclusion-exclusion: 6+6−2=10. Probability = 10/24 = 5/12 ≈ 0.4167. So correct choice from given is none of above; closest is not 1/2. If forced pick, correct fraction 5/12.
Final (correct): 5/12 (≈0.4167).
99. A fair die is rolled 4 times. Probability that at least one 6 appears =
A) 1 − (5/6)^4
B) (1/6)⁴
C) 4(1/6)(5/6)³
D) (5/6)⁴
Answer: A) 1 − (5/6)^4
Explanation: Complement is no 6 in 4 rolls → (5/6)^4.
100. Three people A, B, C each independently flip a fair coin. What is probability that exactly two people get the same result (i.e., two equal, the third different)?
A) 3/4
B) 3/8
C) 3/4? (check)
D) 3/4?
Answer: B) 3/4? Wait compute carefully: Total outcomes = 8. Outcomes where exactly two same and one different: choose which person is the odd one (3 ways) and for each that odd person can be H while the other two T (1) or T while other two H (1) → for each choice 2 outcomes. So total favorable = 3×2=6. Probability = 6/8 = 3/4.
Final Answer: A) 3/4.