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  • Let   and <img alt="\mathrm{B}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B

Let \mathrm{A}  and \mathrm{B} be two events such that \mathrm{P(A \cap B)=\frac{1}{3}, P(A \cup B)=\frac{5}{6}}  and  \mathrm{P(\bar{A})=\frac{1}{2} \cdot}  Then

Option: 1

A, B are independent
 


Option: 2

A, B are mutually exclusive
 


Option: 3

\mathrm{P(A)=P(B)}


Option: 4

\mathrm{P(B) \leq P(A)}


Answers (1)

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\mathrm{\begin{aligned} & P(A \cup B)=P(A)+P(B)-P(A \cap B) \\ & \Rightarrow \quad \frac{5}{6}=\left(1-\frac{1}{2}\right)+P(B)-\frac{1}{3} \Rightarrow P(B)=\frac{2}{3} . \end{aligned}}

\mathrm{P(A) \cdot P(B)=\frac{1}{2} \cdot \frac{2}{3}=\frac{1}{3}=P(A \cap B)}. So, A and B are independent and therefore, not mutually exclusive.

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himanshu.meshram

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