If A and B are any two events such that P A = 2 5 and P A ∩ B = 3 20 , then the conditional…

1. $\frac{11}{20}$
2. $\frac{5}{17}$
3. $\frac{8}{17}$
4. $\frac{1}{4}$

Solution

Given $P\left(A\right)=\frac{2}{5}=\frac{8}{20}$ & $P\left(A\cap B\right)=\frac{3}{20}$

$\therefore P\left(A^{\text{'}}\cup B^{\text{'}}\right)=P{\left(A\cap B\right)}^{\text{'}}$

$=1-P\left(A\cap B\right)$

$=1-\frac{3}{20}=\frac{17}{20}$

$P\left(A\cap \left(A^{\text{'}}\cup B^{\text{'}}\right)\right)=P\left(A\cap B^{\text{'}}\right)$

$=P\left(A\right)-P\left(A\cap B\right)$

$=\frac{8}{20}-\frac{3}{20}=\frac{5}{20}$

$\therefore P\left(A|\left(A^{\text{'}}\cup B^{\text{'}}\right)\right)=\frac{P\left(A\cap \left(A^{\text{'}}\cup B^{\text{'}}\right)\right)}{P\left(A^{\text{'}}\cup B^{\text{'}}\right)}$

$=\frac{\left(\frac{5}{20}\right)}{\left(\frac{17}{20}\right)}=\frac{5}{17}$

Asked in: JEE Main 2016 (09 Apr Online)

Practice more Probability questions on Aicharya