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  • State True or False for the statements in the Exercise. If A and B are independent events, then P(A′ ∪ B) = 1 – P (A) P(B′)

State True or False for the statements in the Exercise.
If A and B are independent events, then P(A′ ∪ B) = 1 – P (A) P(B′)

 

Answers (1)

TRUE

If A and B are independent events, it means that

P(A ∩ B) = P(A)P(B)
P(A′ ∪ B) = P(A’) + P(B) – P(A’ ∩ B)
and P(A′ ∪ B) represents the probability of event ‘only B’ excluding common points.

\\ \therefore P\left(A^{\prime} \cap B\right)=P(B)-P(A \cap B) \\ \Rightarrow P\left(A^{\prime} \cup B\right)=P\left(A^{\prime}\right)+P(B)-P(B)+P(A \cap B) \\ \Rightarrow P\left(A^{\prime} \cup B\right)=1-P(A)+P(A) P(B)\{\text { independent events }\} \\ \Rightarrow P\left(A^{\prime} \cup B\right)=1-P(A)\{1-P(B)\} \\ \Rightarrow P\left(A^{\prime} \cup B\right)=1-P(A) P\left(B^{\prime}\right)

Hence Proved

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infoexpert22

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