Episode 8 — Aptitude and Reasoning / 8.15 — Probability
8.15 Practice MCQs -- Probability
Instructions: Choose the correct option for each question. Detailed solutions are provided after all questions.
Questions
Q1.
A fair die is rolled. What is the probability of getting a number less than 3?
(a) 1/3 (b) 1/2 (c) 1/6 (d) 2/3
Q2.
A card is drawn from a standard deck of 52 cards. What is the probability of drawing a red card?
(a) 1/4 (b) 1/2 (c) 1/13 (d) 3/4
Q3.
A coin is tossed twice. What is the probability of getting at least one head?
(a) 1/2 (b) 1/4 (c) 3/4 (d) 1
Q4.
A bag contains 3 red and 5 blue balls. One ball is drawn. What is the probability it is blue?
(a) 3/8 (b) 5/8 (c) 1/2 (d) 3/5
Q5.
Two dice are thrown. What is the probability of getting a sum of 7?
(a) 1/12 (b) 1/6 (c) 7/36 (d) 5/36
Q6.
A card is drawn from a deck. What is the probability of it being a King or a Queen?
(a) 1/13 (b) 2/13 (c) 1/26 (d) 4/13
Q7.
What is the probability of getting all tails when 3 coins are tossed?
(a) 1/2 (b) 1/4 (c) 1/8 (d) 3/8
Q8.
A die is rolled. What is the probability of NOT getting a 4?
(a) 1/6 (b) 4/6 (c) 5/6 (d) 1/3
Q9.
From a bag with 4 white and 6 black balls, two balls are drawn without replacement. What is the probability that both are black?
(a) 9/25 (b) 1/3 (c) 6/15 (d) 2/5
Q10.
Two dice are thrown. What is the probability of getting a doublet?
(a) 1/36 (b) 1/12 (c) 1/6 (d) 1/3
Q11.
A bag has 5 red and 4 green balls. Two balls are drawn with replacement. What is the probability that both are red?
(a) 25/81 (b) 20/72 (c) 5/18 (d) 10/36
Q12.
The probability of A hitting a target is 1/3 and that of B is 1/5. Both shoot independently. What is the probability that the target is hit?
(a) 7/15 (b) 8/15 (c) 1/15 (d) 2/5
Q13.
Three coins are tossed. What is the probability of getting exactly 2 tails?
(a) 1/8 (b) 1/4 (c) 3/8 (d) 1/2
Q14.
A card is drawn from a deck. What is the probability that it is neither a Heart nor a King?
(a) 9/13 (b) 4/13 (c) 35/52 (d) 36/52
Q15.
Two dice are rolled. What is the probability that the product of the numbers is even?
(a) 1/4 (b) 1/2 (c) 3/4 (d) 2/3
Q16.
A box contains 10 bulbs, 3 of which are defective. If 2 bulbs are selected at random, what is the probability that both are defective?
(a) 1/15 (b) 3/50 (c) 9/100 (d) 1/10
Q17.
The odds against an event are 5:3. What is the probability of the event?
(a) 5/8 (b) 3/8 (c) 3/5 (d) 5/3
Q18.
If P(A) = 0.4, P(B) = 0.5, and P(A and B) = 0.2, find P(A or B).
(a) 0.7 (b) 0.9 (c) 0.6 (d) 0.3
Q19.
A bag has 6 red and 4 blue balls. Three balls are drawn at random. What is the probability that all 3 are red?
(a) 1/6 (b) 1/3 (c) 1/12 (d) 1/2
Q20.
From a deck of 52 cards, two cards are drawn at random. What is the probability that both are Aces?
(a) 1/169 (b) 1/221 (c) 2/52 (d) 1/13
Q21.
A and B are independent events with P(A) = 0.3 and P(B) = 0.4. Find P(A and B).
(a) 0.7 (b) 0.12 (c) 0.1 (d) 0.58
Q22.
If a coin is tossed 5 times, what is the probability of getting at least 4 heads?
(a) 3/16 (b) 1/16 (c) 5/32 (d) 3/32
Q23.
Two dice are thrown. What is the probability that the sum is greater than 10?
(a) 1/12 (b) 1/6 (c) 1/9 (d) 5/36
Q24.
A bag contains 8 red and 5 white balls. 4 balls are drawn. What is the probability that 2 are red and 2 are white?
(a) 140/715 (b) 280/715 (c) 350/715 (d) 420/715
Q25.
A card is drawn from a deck. What is the probability that it is a red King?
(a) 1/52 (b) 1/26 (c) 1/13 (d) 2/13
Q26.
If P(A) = 0.6, what is P(not A)?
(a) 0.6 (b) -0.4 (c) 0.4 (d) 0.5
Q27.
A number is chosen at random from 1 to 20. What is the probability that it is a prime number?
(a) 2/5 (b) 7/20 (c) 1/4 (d) 9/20
Q28.
Three balls are drawn from a bag of 5 red and 3 blue balls. What is the probability of getting all red?
(a) 5/28 (b) 10/56 (c) 5/14 (d) 3/28
Q29.
A die is rolled twice. What is the probability that the sum is 4?
(a) 1/12 (b) 1/9 (c) 1/6 (d) 1/18
Q30.
The probability that it rains on day 1 is 0.3 and on day 2 is 0.4 (independently). What is the probability that it rains on at least one of the two days?
(a) 0.70 (b) 0.58 (c) 0.12 (d) 0.42
Q31.
From 6 boys and 4 girls, 4 children are selected at random. What is the probability that the selection contains exactly 2 boys and 2 girls?
(a) 3/7 (b) 10/21 (c) 90/210 (d) Both (a) and (c)
Q32.
A letter is chosen at random from the word "PROBABILITY". What is the probability that it is a vowel?
(a) 4/11 (b) 3/11 (c) 5/11 (d) 6/11
Q33.
In a single throw of two dice, what is the probability of getting a sum of 11?
(a) 1/36 (b) 1/18 (c) 1/12 (d) 1/9
Q34.
A committee of 3 is chosen from 5 men and 4 women. What is the probability that the committee has at least one woman?
(a) 37/42 (b) 74/84 (c) 5/42 (d) Both (a) and (b)
Q35.
Two cards are drawn from a pack without replacement. What is the probability of getting both from the same suit?
(a) 1/4 (b) 12/51 (c) 4/17 (d) Both (b) and (c)
Q36.
A bag has 3 white and 5 black balls. 4 balls are drawn without replacement. What is the probability of getting exactly 2 white balls?
(a) 15/28 (b) 3/14 (c) 15/70 (d) Both (b) and (c)
Q37.
A biased coin has P(Head) = 1/3. It is tossed 3 times. What is the probability of getting exactly 1 head?
(a) 4/9 (b) 1/3 (c) 2/9 (d) 4/27
Q38.
An urn contains 5 red, 3 green, and 2 blue balls. One ball is drawn. What is the probability that it is not green?
(a) 3/10 (b) 7/10 (c) 1/2 (d) 2/5
Q39.
Two dice are thrown. What is the probability that the numbers shown differ by 2?
(a) 2/9 (b) 1/9 (c) 4/36 (d) 8/36
Q40.
If 4 cards are drawn from a well-shuffled deck of 52, what is the probability that all are from the same suit?
(a) 44/4165 (b) 2860/270725 (c) 1/4 (d) 198/20825
Q41.
A and B take turns throwing a die. A wins if he gets a 6 before B gets a 5. If A starts, what is the probability that A wins?
(a) 6/11 (b) 5/11 (c) 1/2 (d) 36/66
Q42.
Three students A, B, C independently solve a problem with probabilities 1/2, 1/3, and 1/4. What is the probability that the problem is solved?
(a) 1/4 (b) 3/4 (c) 1/24 (d) 23/24
Q43.
A number is selected at random from 1 to 100. What is the probability that it is divisible by both 3 and 7?
(a) 4/100 (b) 5/100 (c) 38/100 (d) 4/99
Q44.
In how many ways can 2 cards drawn from 52 be such that one is red and one is black? What is the probability?
(a) 26/51 (b) 325/1326 (c) 676/1326 (d) 1/2
Answers and Solutions
A1. (a) 1/3
Numbers less than 3: {1, 2} = 2 outcomes
P = 2/6 = 1/3
A2. (b) 1/2
Red cards (Hearts + Diamonds) = 26
P = 26/52 = 1/2
A3. (c) 3/4
P(at least 1 head) = 1 - P(no head) = 1 - P(TT) = 1 - 1/4 = 3/4
A4. (b) 5/8
P(blue) = 5/(3+5) = 5/8
A5. (b) 1/6
Sum 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) = 6 outcomes
P = 6/36 = 1/6
A6. (b) 2/13
Kings = 4, Queens = 4, mutually exclusive.
P = (4+4)/52 = 8/52 = 2/13
A7. (c) 1/8
P(all tails) = (1/2)^3 = 1/8
A8. (c) 5/6
P(not 4) = 1 - 1/6 = 5/6
A9. (b) 1/3
P = 6C2/10C2 = 15/45 = 1/3
A10. (c) 1/6
Doublets = 6 out of 36
P = 6/36 = 1/6
A11. (a) 25/81
With replacement: P(red) = 5/9 each time.
P(both red) = 5/9 x 5/9 = 25/81
A12. (a) 7/15
P(target hit) = 1 - P(both miss)
= 1 - (2/3)(4/5)
= 1 - 8/15
= 7/15
A13. (c) 3/8
P(exactly 2 tails) = 3C2 / 2^3 = 3/8
A14. (d) 36/52
P(Heart or King) = 13/52 + 4/52 - 1/52 = 16/52
P(neither Heart nor King) = 1 - 16/52 = 36/52 = 9/13
Both (a) 9/13 and (d) 36/52 are the same value.
Since 36/52 = 9/13, both are correct. The answer given as option is (d) 36/52.
Actually checking: (a) = 9/13 = 36/52 = (d). Both represent the same probability.
The correct answer is 36/52 = 9/13.
Answer: (d) 36/52, which equals 9/13
A15. (c) 3/4
Product is even if at least one die shows even.
P(product odd) = P(both odd) = (3/6)^2 = 1/4
P(product even) = 1 - 1/4 = 3/4
A16. (a) 1/15
P = 3C2 / 10C2 = 3/45 = 1/15
A17. (b) 3/8
Odds against = 5:3 means unfavourable:favourable = 5:3
P(event) = 3/(5+3) = 3/8
A18. (a) 0.7
P(A or B) = P(A) + P(B) - P(A and B) = 0.4 + 0.5 - 0.2 = 0.7
A19. (a) 1/6
P = 6C3 / 10C3 = 20/120 = 1/6
A20. (b) 1/221
P = 4C2 / 52C2 = 6/1326 = 1/221
A21. (b) 0.12
P(A and B) = P(A) x P(B) = 0.3 x 0.4 = 0.12 (independent)
A22. (d) 3/32
P(at least 4 heads) = P(4H) + P(5H)
= 5C4/32 + 5C5/32
= 5/32 + 1/32
= 6/32
= 3/16
Wait, that gives 3/16 which is option (a).
5C4 = 5, 5C5 = 1
P = (5+1)/32 = 6/32 = 3/16
Answer: (a) 3/16
A23. (a) 1/12
Sum > 10 means sum = 11 or 12.
Sum 11: (5,6),(6,5) = 2
Sum 12: (6,6) = 1
Total = 3
P = 3/36 = 1/12
A24. (b) 280/715
Total = 13C4 = 715
Favourable = 8C2 x 5C2 = 28 x 10 = 280
P = 280/715
A25. (b) 1/26
Red Kings = King of Hearts + King of Diamonds = 2
P = 2/52 = 1/26
A26. (c) 0.4
P(not A) = 1 - P(A) = 1 - 0.6 = 0.4
A27. (a) 2/5
Primes from 1-20: {2, 3, 5, 7, 11, 13, 17, 19} = 8 primes
P = 8/20 = 2/5
A28. (b) 10/56
P = 5C3 / 8C3 = 10/56 = 5/28
Options (a) 5/28 and (b) 10/56 are the same.
Answer: (b) 10/56 = 5/28 = (a). Both are correct.
A29. (a) 1/12
Sum = 4: (1,3),(2,2),(3,1) = 3 outcomes
P = 3/36 = 1/12
A30. (b) 0.58
P(at least one) = 1 - P(none)
= 1 - (0.7)(0.6)
= 1 - 0.42
= 0.58
A31. (d) Both (a) and (c)
Total = 10C4 = 210
Favourable = 6C2 x 4C2 = 15 x 6 = 90
P = 90/210 = 3/7
Both (a) 3/7 and (c) 90/210 are the same value.
A32. (a) 4/11
PROBABILITY: P,R,O,B,A,B,I,L,I,T,Y (11 letters)
Vowels: O, A, I, I = 4 vowels
P = 4/11
A33. (b) 1/18
Sum 11: (5,6),(6,5) = 2 outcomes
P = 2/36 = 1/18
A34. (d) Both (a) and (b)
Total = 9C3 = 84
All men (no women) = 5C3 = 10
At least 1 woman = 84 - 10 = 74
P = 74/84 = 37/42
Both (a) 37/42 and (b) 74/84 are the same.
A35. (d) Both (b) and (c)
Total ways = 52C2 = 1326
Same suit: choose a suit (4 ways), then 2 cards from 13 = 13C2 = 78
Favourable = 4 x 78 = 312
P = 312/1326 = 12/51 = 4/17
Both (b) 12/51 and (c) 4/17 are the same.
A36. (d) Both (b) and (c)
Total = 8C4 = 70
Favourable = 3C2 x 5C2 = 3 x 10 = 30
P = 30/70 = 3/7
Hmm, let me recheck.
3C2 = 3, 5C2 = 10
30/70 = 3/7
But option (b) is 3/14 and (c) is 15/70.
15/70 = 3/14. So (b) and (c) are the same, but my calculation gives 30/70 = 3/7.
Let me recount. 3 white, 5 black, draw 4, exactly 2 white.
Favourable = 3C2 x 5C2 = 3 x 10 = 30
Total = 8C4 = 70
P = 30/70 = 3/7
This doesn't match any individual option perfectly. Let me re-examine.
Actually 3C2 = 3 and 5C2 = 10, giving 30. And 8C4 = 70. So P = 30/70 = 3/7.
None of the given options equal 3/7. Let me re-read the problem.
Oh wait -- option (b) is 3/14 and (c) is 15/70 = 3/14. These equal each other but not 3/7.
Hmm, this seems to be an error in the original options. The correct answer is 3/7. But since the question asks and 30/70 simplifies to 3/7, the closest matching option structure would be that (b) and (c) match each other.
Let me reconsider: maybe I miscounted.
3 white, 5 black, draw 4.
Total = 8C4 = 70. Correct.
Exactly 2 white, 2 black: 3C2 x 5C2 = 3 x 10 = 30. Correct.
P = 30/70 = 3/7.
The correct answer is 3/7 = 30/70. The given options have an error.
Correct answer: 30/70 = 3/7. Note: options (b) and (c) as printed are equal to each other (3/14 = 15/70) but the actual correct probability is 3/7.
A37. (a) 4/9
P(H) = 1/3, P(T) = 2/3, n=3, k=1
P(exactly 1H) = 3C1 x (1/3)^1 x (2/3)^2
= 3 x 1/3 x 4/9
= 12/27
= 4/9
A38. (b) 7/10
P(not green) = 1 - P(green) = 1 - 3/10 = 7/10
A39. (a) 2/9
Pairs differing by 2:
(1,3),(3,1),(2,4),(4,2),(3,5),(5,3),(4,6),(6,4) = 8 outcomes
P = 8/36 = 2/9
A40. (a) 44/4165
Total = 52C4 = 270,725
Same suit: 4 x 13C4 = 4 x 715 = 2,860
P = 2,860 / 270,725 = 44/4165
(Simplified by dividing numerator and denominator by 65: 2860/65 = 44, 270725/65 = 4165)
A41. (a) 6/11
P(A gets 6) = 1/6, P(A misses) = 5/6
P(B gets 5) = 1/6, P(B misses) = 5/6
A wins on 1st turn: 1/6
A wins on 2nd round: (5/6)(5/6)(1/6) = 25/216
...
This is a geometric series.
P(A wins) = (1/6) / [1 - (5/6)(5/6)]
= (1/6) / [1 - 25/36]
= (1/6) / (11/36)
= 36/66
= 6/11
A42. (b) 3/4
P(none solves) = (1/2)(2/3)(3/4) = 6/24 = 1/4
P(problem solved) = 1 - 1/4 = 3/4
A43. (a) 4/100
Divisible by both 3 and 7 = divisible by 21.
Multiples of 21 from 1-100: 21, 42, 63, 84 = 4 numbers
P = 4/100 = 1/25
A44. (a) 26/51
Total ways = 52C2 = 1,326
One red, one black = 26C1 x 26C1 = 676
P = 676/1326 = 26/51
Option (c) 676/1326 is the same as (a) 26/51.
But since the question asks for the probability, the simplified form is 26/51.
Score Guide
40-44 correct: Excellent -- you have mastered Probability
32-39 correct: Good -- review the concepts you missed
24-31 correct: Average -- revisit the formulas and practice more
Below 24: Needs work -- study 8.15.a thoroughly before retrying
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