A) \[P(A\cup B)=\frac{5}{3}\]
B) \[P\left( \frac{A}{B} \right)=\frac{1}{3}\]
C) \[P\left( \frac{A}{A\cup B} \right)=\frac{5}{6}\]
D) All of these
Correct Answer: C
Solution :
Since, \[A\] and \[B\] are independent events \[\therefore \] \[P(A\cap B)=P(A)P(B)\] \[=\frac{1}{2}\cdot \frac{1}{5}=\frac{1}{10}\] Now,\[P(A\cup B)=P(A)+P(B)-P(A\cap B)\] \[\Rightarrow \] \[P(A\cup B)=\frac{1}{2}+\frac{1}{5}-\frac{1}{10}\] \[\Rightarrow \] \[P(A\cup B)=\frac{5+2-1}{10}=\frac{6}{10}=\frac{3}{5}\] and \[P\left( \frac{A}{B} \right)=P(A)\] (since, \[A\] and \[B\] are independent) \[\Rightarrow \] \[P\left( \frac{A}{B} \right)=\frac{1}{2}\] \[\left[ \because \,\,P(A)=\frac{1}{2} \right]\] and\[P\left( \frac{A}{A\cup B} \right)=\frac{P[A\cap (A\cup B)]}{P(A\cup B)}=\frac{P(A)}{P(A\cup B)}\] \[P\left( \frac{A}{A\cup B} \right)=\frac{\frac{1}{2}}{\frac{3}{5}}=\frac{5}{6}\]You need to login to perform this action.
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