Mathematics - Probability Question with Solution | TestHub

$\frac{P\left(E_1\cap E_2\right)}{P\left(E_2\right)}=\frac{1}{2}, \frac{P\left(E_2\cap E_1\right)}{P\left(E_1\right)}=\frac{3}{4}, P\left(E_1\cap E_2\right)=\frac{1}{8}$

$\Rightarrow P\left(E_2\right)=\frac{1}{4}, P\left(E_1\right)=\frac{1}{6}, P\left(E_1\cup E_2\right)=\frac{1}{6}+\frac{1}{4}-\frac{1}{8}=\frac{7}{24}$

$P\left(E^{\text{'}}\cap E_2^{\text{'}}\right)=P{\left(E_1\cup E_2\right)}^{\text{'}}=1-\frac{7}{24}=\frac{17}{24}$

$P\left(E_2^{\text{'}}\right)=\frac{3}{4} \Rightarrow P\left(E_1\cap E_2\right)=P\left(E_1\right)P\left(E_2^{\text{'}}\right)$

$P\left(E_1\cap E_2^{\text{'}}\right)=P\left(E_1\right)-P\left(E_1\cap E_2\right)=\frac{1}{6}-\frac{1}{8}=\frac{1}{24}$

$P\left(E_1\cap E_2^{\text{'}}\right)=P\left(E_1\right)·P\left(E_2\right)$