# Probability and Statistics Questions and Answers – Set Theory of Probability – 1

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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

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 12 and B not solving it is 14. The whole probability of the problem being solved is 6364 then what is the probability of solving it?
a) 18
b) 164
c) 78
d) 12

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) = 12, P(~B) = 14 and P(A ∪ B ∪ C) = 63/64

We know P(A ∪ B ∪ C) = 1 – P(A ∪ B ∪ C)

= 1 – P(ABC)

= 1 – P(A) P(B) P(C)

Let P(C) = p
ie 6364 = 1 – (12)(14)(p)

= 1 – p8
⇒ P =1/8 = P(C)
⇒P(C) = 1 – P = 1 – 18 = 78.

3. Let A and B be two events such that P(A) = 15 While P(A or B) = 12. Let P(B) = P. For what values of P are A and B independent?
a) 110 and 310
b) 310 and 45
c) 38 only
d) 310

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)
= 15 + P (15)P
12 = 15 + 45P
⇒ P= 38.

4. If A and B are two mutually exclusive events with P(~A) = 56 and P(b) = 13 then P(A /~B) is equal to ___________
a) 14
b) 12
c) 0, since mutually exclusive
d) 518

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) 13

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

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)

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

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)

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)

Explanation: A ⊂ B => BA
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)

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

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

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)=$$\sum_{i=1}^n P(A_i)$$

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