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Events A and B are such that P(A)=1 / 2, P(B)=7 / 12 and P( not A or n o t B)=1 / 4. State whether A and B are independent?
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Events A and B are such that P(A)=1 / 2, P(B)=7 / 12 and P( not A or n o t B)=1 / 4. State whether A and B are independent?

Solution:

Given: P(A)=1 / 2, P(B)=7 / 12 and P( not A or \operatorname{not} B)=1 / 4
Concept: Two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of occurrence of the other.

Calculation:
Evaluating the value of required parameter

\Rightarrow P\left(A^{\prime} \cup B^{\prime}\right)=1 / 4

\Rightarrow P(A \cap B)^{\prime}=1 / 4

\Rightarrow 1-P(A \cap B)=1 / 4

\Rightarrow P(A \cap B)=1-1 / 4

\Rightarrow P(A \cap B)=3 / 4 \ldots \ldots .(1) .

And P (A). P(B)=1 / 2 \times 7 / 12=7 / 24 \ldots . . (2)

From (1) and (2) P(A \cap B) \neq P (A). P(B)

Final Answer: Therefore, A and B are not independent events.

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