Question

A and B are two events such that \(P(A)=0.54,P(B)=0.69 \text{ and } P(A∩B)=0.35\). Find\((i)P(A∪B)(ii)P(A´∩B´)(iii)P(A∩B´)(iv)P(B∩A´)\)

Answer 1

Given:

P(A) = 0.54
P(B) = 0.69
P(A ∩ B) = 0.35

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(i) Find P(A ∪ B)

Using the formula:

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

P(A ∪ B) = 0.54 + 0.69 − 0.35

P(A ∪ B) = 0.88

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(ii) Find P(A′ ∩ B′)

Using the relation:

P(A′ ∩ B′) = 1 − P(A ∪ B)

P(A′ ∩ B′) = 1 − 0.88

P(A′ ∩ B′) = 0.12

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(iii) Find P(A ∩ B′)

Using the relation:

P(A ∩ B′) = P(A) − P(A ∩ B)

P(A ∩ B′) = 0.54 − 0.35

P(A ∩ B′) = 0.19

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(iv) Find P(B ∩ A′)

Using the relation:

P(B ∩ A′) = P(B) − P(A ∩ B)

P(B ∩ A′) = 0.69 − 0.35

P(B ∩ A′) = 0.34

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Final Answers:

(i) P(A ∪ B) = 0.88
(ii) P(A′ ∩ B′) = 0.12
(iii) P(A ∩ B′) = 0.19
(iv) P(B ∩ A′) = 0.34