A bag contains four tickets marked with numbers 112, 121, 211, 222. One ticket is drawn at random from the bag. Let Ei (i = 1, 2, 3) denote the event that ith digit on the selected ticket is 2, then which of the following is not correct ?
Answer & explanation
Correct answer: option 4
$P\left(E_1\right)=\frac{2}{4}=\frac{1}{2}, P\left(E_2\right)=\frac{2}{4}=\frac{1}{2}, P\left(E_3\right)=\frac{2}{4}=\frac{1}{2}$
$P\left(E_1 \cap E_2\right)=\frac{1}{4}=P\left(E_1\right) .\left(E_2\right)$
$\Rightarrow$ '$E_1$' and '$E_2$' are independent.
$P\left(E_1 \cap E_3\right)=\frac{1}{4}=P\left(E_1\right) .\left(E_3\right)$
$\Rightarrow$ '$E_1$' and '$E_3$' are independent.
$P\left(E_2 \cap E_3\right)=\frac{1}{4}=P\left(E_2\right) .\left(E_3\right)$
$\Rightarrow$ '$E_2$' and '$E_3$' are independent.
$P\left(E_1 \cap E_2 \cap E_3\right)=\frac{1}{4} \neq P\left(E_1\right) . P\left(E_2\right) .\left(E_3\right)$
$\Rightarrow E_1, E_2$ and $E_3$ are not independent.