20 mcqs numericals with 4 options on probability only with explained answers
Here is a set of 20 multiple-choice numerical questions on probability covering Multiplication Theorem, Conditional Probability, Independent Events, Total Probability, Baye’s Theorem, and Partition of a Sample Space, with explained answers:
1. Two dice are rolled. What is the probability that the sum is 8?
a) 5/36
b) 1/6
c) 7/36
d) 1/12
Explanation: Possible pairs are (2,6),(3,5),(4,4),(5,3),(6,2) → 5 ways; total=36.
2. If $$P(A) = \frac{1}{3}$$, $$P(B) = \frac{1}{4}$$ and $$A,B$$ independent, $$P(A \cap B)$$ = ?
a) 1/12
b) 1/7
c) 1/6
d) 1/9
Explanation: For independent events $$P(A \cap B) = P(A)P(B) = \frac{1}{3} \times \frac{1}{4} = \frac{1}{12}$$.
3. A bag has 3 white and 2 black balls. Two balls drawn without replacement. Probability both are white?
a) 3/10
b) 1/2
c) 3/5
d) 1/5
Explanation: $$ \frac{3}{5} \times \frac{2}{4} = \frac{6}{20} = \frac{3}{10} $$.
4. If $$P(A|B) = \frac{1}{2}$$, $$P(B) = \frac{1}{3}$$. Find $$P(A \cap B)$$?
a) 1/6
b) 1/2
c) 1/3
d) 1/5
Explanation: $$P(A \cap B) = P(A|B)P(B) = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$$.
5. Two cards drawn from a deck. Probability both are kings (without replacement)?
a) 1/221
b) 1/169
c) 1/52
d) 1/325
Explanation: $$ \frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}$$.
6. Events $$A$$ and $$B$$ are such that $$P(A) = 0.4$$, $$P(B) = 0.5$$, $$P(A \cup B) = 0.7$$. Find $$P(A \cap B)$$?
a) 0.2
b) 0.3
c) 0.4
d) 0.1
Explanation: $$P(A \cap B) = P(A) + P(B) - P(A \cup B) = 0.4 + 0.5 - 0.7 = 0.2$$.
7. If $$A$$ and $$B$$ are independent and $$P(B) = 0.6$$, $$P(A \cap B) = 0.18$$, find $$P(A)$$?
a) 0.3
b) 0.4
c) 0.5
d) 0.6
Explanation: $$P(A) = \frac{P(A \cap B)}{P(B)} = \frac{0.18}{0.6} = 0.3$$.
8. A box has 2 red, 3 green, 5 blue balls. If one ball drawn, $$P($$green or blue$$) = ?$$
a) 1/2
b) 4/5
c) 3/5
d) 1/5
Explanation: $$P(\text{green or blue}) = \frac{3+5}{10} = \frac{8}{10} = \frac{4}{5}$$.
9. Three machines produce 30%, 50%, and 20% of bulbs. Their defective rates are 2%, 3%, and 5%. Find probability a random bulb is defective.
a) 0.034
b) 0.03
c) 0.04
d) 0.05
Explanation: Total defective $$= 0.3 \times 0.02 + 0.5 \times 0.03 + 0.2 \times 0.05 = 0.034$$.
10. Using above data, if a bulb is defective, find probability it’s from second machine.
a) 0.44
b) 0.5
c) 0.38
d) 0.6
Explanation: Use Baye’s Theorem: $$\frac{0.5 \times 0.03}{0.034} = 0.44$$.
11. If $$P(A) = \frac{1}{2}$$, $$P(B|A) = \frac{1}{3}$$, find $$P(A \cap B)$$.
a) 1/6
b) 1/5
c) 1/3
d) 1/2
Explanation: $$P(A \cap B)=P(A) \times P(B|A) = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$$.
12. Events A and B are mutually exclusive. If $$P(A) = 0.3$$, $$P(B) = 0.4$$, $$P(A \cup B)$$ = ?
a) 0.7
b) 0.1
c) 0.12
d) 0.0
Explanation: For mutually exclusive, $$P(A \cup B) = P(A) + P(B) = 0.7$$.
13. Probability of drawing face card from deck is?
a) 3/13
b) 3/52
c) 12/52
d) 1/4
Explanation: Face cards = 12, total cards 52, so $$12/52 = 3/13$$.
14. Two dice rolled, probability that one shows 4 and other shows 5.
a) 1/18
b) 1/36
c) 1/12
d) 1/9
Explanation: Two outcomes (4,5) or (5,4), total 36; probability = $$2/36 = 1/18$$.
15. In a box, probability of red ball is 0.3, green is 0.5, blue is 0.2. What is probability ball drawn is not green?
a) 0.3
b) 0.5
c) 0.7
d) 0.2
Explanation: Not green = 1 - 0.5 = 0.5.
16. A card drawn from a pack is an ace or a king. Probability?
a) 2/13
b) 1/13
c) 1/26
d) 4/52
Explanation: Total aces + kings = 8 cards; $$8/52 = 2/13$$.
17. Two events A and B are independent, $$P(A) = 0.3$$, $$P(B) = 0.4$$. Find $$P(A' \cap B)$$.
a) 0.28
b) 0.42
c) 0.12
d) 0.1
Explanation: $$P(A') = 0.7$$, $$P(A' \cap B) = P(A') \times P(B) = 0.7 \times 0.4 = 0.28$$.
18. Probability of rain on any day is 0.2. Find probability it rains on exactly one day out of next 2 days.
a) 0.32
b) 0.16
c) 0.36
d) 0.4
Explanation: $$P(\text{rain on exactly 1 day}) = 2 \times 0.2 \times 0.8 = 0.32$$.
19. If $$P(A) = 0.7$$, $$P(B|A) = 0.4$$, find $$P(A \cap B')$$.
a) 0.42
b) 0.28
c) 0.3
d) 0.18
Explanation: $$P(A \cap B') = P(A) \times (1-P(B|A)) = 0.7 \times 0.6 = 0.42$$.
20. Sample space partitioned into 3 events $$A_1, A_2, A_3$$ with probabilities 0.2, 0.5, 0.3. $$P(B|A_1) = 0.1$$, $$P(B|A_2) = 0.4$$, $$P(B|A_3) = 0.7$$. $$P(B) =$$ ?
a) 0.43
b) 0.4
c) 0.39
d) 0.41
Explanation: Use total probability: $$0.2 \times 0.1 + 0.5 \times 0.4 + 0.3 \times 0.7 = 0.43$$.
21 to 40
Here are 20 more multiple-choice numerical questions on probability (21 to 40) covering deeper concepts related to Multiplication Theorem, Conditional Probability, Independent Events, Total Probability, Baye’s Theorem, and Partition of a Sample Space, with explained answers:
21. A box contains 5 white and 3 black balls. If two balls are drawn with replacement, probability both are black?
a) 9/64
b) 1/4
c) 3/8
d) 1/8
Explanation: $$P = \frac{3}{8} \times \frac{3}{8} = \frac{9}{64}$$.
22. If $$P(A) = 0.5$$, $$P(B) = 0.7$$, and $$P(A \cap B) = 0.35$$, are $$A$$ and $$B$$ independent?
a) Yes
b) No
c) Cannot say
d) None
Explanation: Since $$0.5 \times 0.7 = 0.35 = P(A \cap B)$$, they are independent.
23. A die is rolled. What is the probability of getting an even number given the number is greater than 3?
a) 1/3
b) 1/2
c) 2/3
d) 1/6
Explanation: Numbers >3 are 4,5,6; even numbers are 4,6, so $$P=2/3$$.
24. If probability of rain on Monday is 0.3 and on Tuesday is 0.4, independent of Monday, probability it rains both days?
a) 0.12
b) 0.7
c) 0.1
d) 0.3
Explanation: Independent: multiply $$0.3 \times 0.4 = 0.12$$.
25. A card is drawn. Probability it is a queen or heart?
a) 4/13
b) 17/52
c) 1/4
d) 1/13
Explanation: $$P(Q) = 4/52$$, $$P(H) = 13/52$$, $$P(Q \cap H) = 1/52$$. Total = $$\frac{4+13-1}{52} = \frac{16}{52}=\frac{4}{13}$$.
26. Three machines A, B, C produce 40%, 35%, 25%. Defect rates 3%, 4%, 5%. Probability a defect is from A?
a) 0.49
b) 0.48
c) 0.43
d) 0.50
Explanation: Use Bayes: $$P(A|D) = \frac{0.4 \times 0.03}{0.4 \times 0.03 + 0.35 \times 0.04 + 0.25 \times 0.05} = 0.49$$.
27. Events A and B such that $$P(A) = 0.6$$, $$P(B) = 0.5$$, $$P(A \cap B) = 0.3$$, find $$P(A \cup B')$$.
a) 0.8
b) 0.9
c) 0.7
d) 0.6
Explanation: $$P(A \cup B') = 1 - P(A^c \cap B) = 1 - (1-0.6) \times 0.5 = 1 - 0.2 = 0.8$$.
28. Probability of drawing a red card from deck is?
a) 1/2
b) 1/4
c) 1/3
d) 1/13
Explanation: Half cards are red (26 out of 52), so 1/2.
29. Two coins tossed. Probability at least one head?
a) 1/2
b) 3/4
c) 1/4
d) 1/3
Explanation: Only (T,T) fails, so $$1 - 1/4 = 3/4$$.
30. If $$P(A) = 0.7$$, $$P(B) = 0.5$$, find $$P(A \cap B^c)$$ if independent.
a) 0.35
b) 0.2
c) 0.3
d) 0.4
Explanation: $$P(B^c) = 0.5$$, so $$P(A \cap B^c) = P(A) \times P(B^c) = 0.7 \times 0.5 = 0.35$$.
31. Probability of defective bulb from 3 machines with defect rates 1%, 2%, 4%, and proportions 0.2, 0.5, 0.3?
a) 0.023
b) 0.02
c) 0.025
d) 0.028
Explanation: $$0.2*0.01 + 0.5*0.02 + 0.3*0.04 = 0.023$$.
32. Two events A and B mutually exclusive with $$P(A) = 0.3$$, $$P(B) = 0.4$$. $$P(A \cap B) = ?$$
a) 0
b) 0.7
c) 0.12
d) 0.1
Explanation: 0 for mutually exclusive events.
33. Two dice rolled. Probability sum is either 7 or 11?
a) 8/36
b) 7/36
c) 2/9
d) 9/36
Explanation: Sum 7: 6 ways; sum 11: 2 ways; total 8/36.
34. Probability that an event occurs at least once in 3 independent trials with $$p=0.2$$?
a) 0.488
b) 0.6
c) 0.3
d) 0.512
Explanation: $$1 - (1-0.2)^3 = 0.488$$.
35. Probability of drawing ace or heart from deck?
a) 4/13
b) 1/4
c) 1/13
d) 5/13
Explanation: $$4$$ aces $$+ 13$$ hearts $$-$$ 1 ace of hearts = $$4+13-1=16$$, so $$16/52=4/13$$.
36. Probability of getting a 5 or 6 on a die roll?
a) 1/3
b) 1/6
c) 1/2
d) 1/4
Explanation: 2 favorable outcomes, probability $$2/6 = 1/3$$.
37. If $$P(A) = 0.8$$, $$P(B|A) = 0.1$$, find $$P(A^c \cup B)$$.
a) 0.82
b) 0.18
c) 0.26
d) 0.72
Explanation: $$P(A^c \cup B) = 1 - P(A \cap B^c) = 1 - P(A)(1-P(B|A)) = 1-0.8 \times 0.9=0.28$$.
38. A box contains 4 defective out of 10 bulbs. 2 bulbs drawn with replacement. Probability none defective?
a) 0.36
b) 0.6
c) 0.16
d) 0.4
Explanation: $$P(non-defective)=6/10$$, so $$ (6/10)^2=0.36$$.
39. Probability of selecting a queen or a red card from deck?
a) 4/13
b) 1/2
c) 1/4
d) 3/13
Explanation: Same as question 25, answer $$4/13$$.
40. Probability that a randomly chosen card is king given it is a face card?
a) 1/3
b) 1/4
c) 1/2
d) 2/3
Explanation: 12 face cards, 4 kings, so $$4/12 = 1/3$$.
