In an experimental performance of a single throw of a pair of unbiased normal dice, let three events  $E_1,E_{2}and{E}_3$  are defined as follows 

$E_1:$  getting prime numbered face on each dice.

 $E_2:$ getting the same number on each dice

$E_3:$  getting total on two dice equal to 4.

Which of the following is/are true ?

$$\begin{gathered} E_1=\{(2,2),(2,3),(3,2),(2,5),(5,2)(3,5),(5,3),(3,3).(5,5)\} \\ {E}_2=\{(1,1),(2,2),(3,3),(4,4),(5,5)(6,6)\} \\ {E}_3=\{(1,3),(3,1),(2,2)\} \\ P\left(E_1\right)=\frac{9}{36};P\left(E_2\right)=\frac{6}{36};P\left(E_3\right)=\frac{3}{36} \\ Clearly, P\left(E_1\right),P\left(E_2\right),P\left(E_3\right) are in A.P. \Rightarrow \left(a\right) is correct \\ P(E_1\cap E_2)=\frac{3}{36}=\frac{1}{12}\ne P\left(E_1\right)P\left(E_2\right)\Rightarrow \left(b\right) is incorrect \\ Now, P(E_3/E_1)=\frac{P(E_3\cap E_1)}{P\left(E_1\right)}=\frac{1}{9}\Rightarrow \left(c\right) is incorrect \\ Also, P(E_1+E_2)+P(E_2-E_3)=[P\left(E_1\right)+P\left(E_2\right)-P(E_1\cap E_2)] +[P\left(E_2\right)-P(E_1\cap E_3)] =\frac{9}{36}+\frac{12}{36}-\frac{3}{36}-\frac{1}{36}=\frac{17}{36} \\ \left(d\right) is correct \end{gathered}$$