Mathematics - Probability Question with Solution | TestHub

$E_1\to A$show up$4$

$E_2\to B$shows up$2$

$E_3\to$Sum is odd (i.e., even$+$odd or odd$+$even)

$P\left(E_1\right)=\frac{6}{6.6}=\frac{1}{6}$

$P\left(E_2\right)=\frac{6}{6.6}=\frac{1}{6}$

$P\left(E_3\right)=$$\frac{3.3}{6.6}+\frac{3.3}{6.6}=\frac{2}{4}=\frac{1}{2}$

$P\left(E_1\cap E_2\right)=\frac{1}{6.6}=P\left(E_1\right) . P\left(E_2\right)$

$\Rightarrow$$E_1 \& E_2$are independent

$P\left(E_1\cap E_3\right)=\frac{1.3}{6.6}=P\left(E_1\right) . P\left(E_3\right)$

$\Rightarrow$$E_1 \& E_3$are independent

$P\left(E_2\cap E_3\right)=\frac{1.3}{6.6}=\frac{1}{12}=P\left(E_2\right) . P\left(E_3\right)$

$\Rightarrow$$E_2 \& E_3$are independent

$P\left(E_1\cap E_2\cap E_3\right)=0$$\ne P\left(E_1\right)\times P\left(E_2\right)\times P\left(E_3\right)$