Let two fair six-faced dice $A$ and $B$ be thrown simultaneously. If $E_1$ is the event that die $A$ shows up four, $E_2$ is the event that die $B$ shows up two and $E_3$ is the event that the sum of numbers on both dice is odd, then which of the following statements is NOT true ?
- $E_1$ and $E_3$ are independent.
- $E_1$ , $E_2$ and $E_3$ are independent
- $E_1$ and $E_2$ are independent.
- $E_2$ and $E_3$ are independent.
My attempt :
$E_1=(4,1) (4,2) (4,3) (4,4) (4,5) (4,6),$
$E_2 = (1,2) (2,2) (3,2) (4,2) (5,2) (6,2)$
$E_3 = (1,2)....... (6,5)$ One even and one odd
$P(E_2) = 1/6$
$P(E_1.E_2.E_3) = 0$
$P(E_1.E_2) = 1/36 //(4,2)$
$P(E_1).P(E_2) =1/6.1/6 =1/36$
$E_1 ,E_2$ are dependent.
But somewhere, answer is given option $(2)$.
Can you explain in formal way, please?