11th Class Mathematics Sample Paper Mathematics Olympiad - Sample Paper-4

  • question_answer If A and B are two independent events such that \[\mathbf{P}\left( A \right)=\frac{3}{10},\mathbf{P}\left( \mathbf{B}' \right)=\mathbf{x}\]and \[\mathbf{P}\left( \mathbf{A}\cup \mathbf{B} \right)=\frac{8}{10}\], then x is equal to

    A) \[\frac{2}{7}\]                          

    B) \[\frac{3}{7}\]               

    C) \[\frac{4}{7}\]                          

    D) \[\frac{5}{7}\]

    Correct Answer: A

    Solution :

    [a]   \[P\left( A\cup B \right)=\frac{8}{10}\] \[P\left( B' \right)=x\Rightarrow P\left( B \right)=1-x\] \[P(A)=\frac{3}{10}\] Now \[A\And B\] be independent. \[P\left( A\cap B \right)=P\left( A \right).P\left( B \right)=\left( 1-x \right).\frac{3}{10}\] \[\because P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)-P\left( A\cap B \right)\] \[\frac{8}{10}=\frac{3}{10}+(1-x)-\frac{-3(1-x)}{10}\] \[\Rightarrow \frac{8}{10}=\frac{3+10\,(1-x)-3\,(1-x)}{10}\,\Rightarrow 8=3+7(1-x)\] \[\Rightarrow 8=3+7-7x\Rightarrow 7x=10-8=2\Rightarrow x=\frac{2}{7}.\]Hence, Option [a]is correct.

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