This set of Probability and Statistics Multiple Choice Questions & Answers (MCQs) focuses on “Set Theory of Probability – 1”.
1. A and B are two events such that P(A) = 0.4 and P(A ∩ B) = 0.2 Then P(A ∩ B) is equal to ___________
a) 0.4
b) 0.2
c) 0.6
d) 0.8
View Answer
Explanation: P(A ∩ B) = P(A – (A ∩ B))
= P(A) – P(A ∩ B)
= 0.6 – 0.2 Using P(A) = 1 – P(A)
= 0.4.
2. A problem in mathematics is given to three students A, B and C. If the probability of A solving the problem is 1⁄2 and B not solving it is 1⁄4. The whole probability of the problem being solved is 63⁄64 then what is the probability of solving it?
a) 1⁄8
b) 1⁄64
c) 7⁄8
d) 1⁄2
View Answer
Explanation:
Let A be the event of A solving the problem
Let B be the event of B solving the problem
Let C be the event of C solving the problem
Given P(a) = 1⁄2, P(~B) = 1⁄4 and P(A ∪ B ∪ C) = 63/64
We know P(A ∪ B ∪ C) = 1 – P(A ∪ B ∪ C)
= 1 – P(A ∩ B ∩ C)
= 1 – P(A) P(B) P(C)
Let P(C) = p
ie 63⁄64 = 1 – (1⁄2)(1⁄4)(p)
= 1 – p⁄8
⇒ P =1/8 = P(C)
⇒P(C) = 1 – P = 1 – 1⁄8 = 7⁄8.
3. Let A and B be two events such that P(A) = 1⁄5 While P(A or B) = 1⁄2. Let P(B) = P. For what values of P are A and B independent?
a) 1⁄10 and 3⁄10
b) 3⁄10 and 4⁄5
c) 3⁄8 only
d) 3⁄10
view Answer
Explanation: For independent events,
P(A ∩ B) = P(A) P(B)
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
= P(A) + P(B) – P(A) P(B)
= 1⁄5 + P (1⁄5)P
⇒ 1⁄2 = 1⁄5 + 4⁄5P
⇒ P= 3⁄8.
4. If A and B are two mutually exclusive events with P(~A) = 5⁄6 and P(b) = 1⁄3 then P(A /~B) is equal to ___________
a) 1⁄4
b) 1⁄2
c) 0, since mutually exclusive
d) 5⁄18
View Answer
Explanation: As A and B are mutually exclusive we have
\(A\cap\bar{B}\)
And Hence
\(P(A/\bar{B})=\frac{P(A\cap\bar{B})}{P(\bar{B})}\)
\(\frac{1-P(\bar{A})}{1-P(\bar{B})}=\frac{1-\frac{5}{6}}{1-\frac{1}{3}}\)
\(P(A/\bar{B})=\frac{1}{4}\)
5. If A and B are two events such that P(a) = 0.2, P(b) = 0.6 and P(A /B) = 0.2 then the value of P(A /~B) is ___________
a) 0.2
b) 0.5
c) 0.8
d) 1⁄3
View Answer
Explanation: For independent events,
P(A /~B) = P(a) = 0.2.
6. If A and B are two mutually exclusive events with P(a) > 0 and P(b) > 0 then it implies they are also independent.
a) True
b) False
View Answer
Explanation: P(A ∩ B) = 0 as (A ∩ B) = ∅
But P(A ∩ B) ≠ 0 , as P(a) > 0 and P(b) > 0
P(A ∩ B) = P(A) P(B), for independent events.
7. Let A and B be two events such that the occurrence of A implies occurrence of B, But not vice-versa, then the correct relation between P(a) and P(b) is?
a) P(A) < P(B)
b) P(B) ≥ P(A)
c) P(A) = P(B)
d) P(A) ≥ P(B)
View Answer
Explanation: Here, according to the given statement A ⊆ B
P(B) = P(A ∪ (A ∩ B)) (∵ A ∩ B = A)
= P(A) + P(A ∩ B)
Therefore, P(B) ≥ P(A)
8. In a sample space S, if P(a) = 0, then A is independent of any other event.
a) True
b) False
View Answer
Explanation: P(a) = 0 (impossible event)
Hence, A is not dependent on any other event.
9. If A ⊂ B and B ⊂ A then,
a) P(A) > P(B)
b) P(A) < P(B)
c) P(A) = P(B)
d) P(A) < P(B)
View Answer
Explanation: A ⊂ B and B ⊂ A => A = B
Hence P(a) = P(b).
10. If A ⊂ B then?
a) P(a) > P(b)
b) P(A) ≥ P(B)
c) P(B) = P(A)
d) P(B) = P(B)
View Answer
Explanation: A ⊂ B => B ⊂ A
Therefore, P(A) ≥ P(B)
11. If A is a perfect subset of B and P(a < Pb), then P(B – A) is equal to ____________
a) P(a) / P(b)
b) P(a)P(b)
c) P(a) + P(b)
d) P(b) – P(a)
View Answer
Explanation: From Basic Theorem of probability,
P(B – A) = P(b) – P(a), this is true only if the condition given in the question is true.
12. What is the probability of an impossible event?
a) 0
b) 1
c) Not defined
d) Insufficient data
View Answer
Explanation: If the probability of an event is 0, then it is called as an impossible event.
13. If A = A1 ∪ A2……..∪ An, where A1…An are mutually exclusive events then?
a) \(\sum_{i=0}^n P(A_i)\)
b) \(\sum_{i=1}^n P(A_i)\)
c) \(\prod_{i=0}^n P(A_i)\)
d) Not defined
View Answer
Explanation: A = A1 ∪ A2……..∪ An, where A1…An
Since A1…An are mutually exclusive
P(a) = P(A1) + P(A2) + … + P(An)
Therefore p(a)=\(\sum_{i=1}^n P(A_i)\)
Sanfoundry Global Education & Learning Series – Probability and Statistics.
To practice all areas of Probability and Statistics, here is complete set of 1000+ Multiple Choice Questions and Answers.
Participate in the Sanfoundry Certification contest to get free Certificate of Merit. Join our social networks below and stay updated with latest contests, videos, internships and jobs!
