Q103 / 180 · COMEDK 2014 · Mathematics

Given that, $P(A)=0.1, P(B \mid A)=0.6$ and $P\left(B \mid A^{C}\right)=0.3$ what is $P(A \mid B)=$ ?

  • Correct answer
    A
    $\frac{2}{11}$
  • B
    $\frac{4}{11}$
  • C
    $\frac{7}{11}$
  • D
    $\frac{9}{11}$

Solution

Given, $P(B \mid A)=0.6$ $\begin{array}{llrl}\Rightarrow & & \frac{P(B \cap A)}{P(A)}= & =0.6 \\ \Rightarrow & & P(B \cap A)=0.6 \times 0.1=0.06 \\ \text { Also, } & & \frac{P\left(B \mid A^{C}\right)}{}=0.3 \\ \Rightarrow & & \frac{P\left(B \cap A^{C}\right)}{P\left(A^{C}\right)}=0.3 \\ \Rightarrow & & \frac{P(B)-P(B \cap A)}{1-P(A)} & =0.3 \\ \Rightarrow & & \frac{P(B)-0.06}{0.9} & =0.3 \\ \Rightarrow & & P(B) & =0.33\end{array}$ Now, $P(A \mid B)=\frac{P(A \cap B)}{P(B)}=\frac{0.06}{0.33}=\frac{2}{11}$

Hence, the correct answer is option: 'A'.

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