Question

A bag contains four tickets with numbers $221,212,122,111$, One ticket is drawn at random from the bag. Let $E_i(i=1,2,3)$ denote the event that the it digit on the ticket is $1$. Then

• A. $E_1$ and $E_2$ are independent
• B. $E_2$ and $E_3$ are independent
• C. $E_1$ and $E_3$ are independent
• D. all the above

Answer 1

The correct option is D all the above

$P\left(E_1\right)=\frac{1}{2},P\left(E_2\right)=\frac{1}{2},P\left(E_3\right)=\frac{1}{2}$
$P(E_1\cap E_2)=\frac{1}{4}=P\left(E_1\right)\times P\left(E_2\right)$ Hence, $E_1$ and $E_2$ are independent.
$P(E_2\cap E_3)=\frac{1}{4}=P\left(E_2\right)\times P\left(E_3\right)$ Hence, $E_2$ and $E_3$ are independent.
$P(E_1\cap E_3)=\frac{1}{4}=P\left(E_1\right)\times P\left(E_3\right)$ Hence, $E_1$ and $E_3$ are independent.