Question

A bag contains four tickets marked with 112, 121, 211, 222. One ticket is drawn at random from the bag. Let $E_i(i=1,2,3)$ denote the event that $i^{th}$ digit on the ticket is 2. Then

• A. $E_1$ and $E_2$ are independent
• B. $E_2$ and $E_3$ are independent
• C. $E_3$ and $E_1$ are independent
• D. $E_1,E_2$ and $E_3$ are independent

Answer 1

Answer: (A), (B), (C)

$E_1$ and $E_2$ are independent
$E_2$ and $E_3$ are independent
$E_3$ and $E_1$ are independent

Events $A$ and $B$ are independent if $P\left(A\right)\bigcap P\left(B\right)=P\left(A\right)\times P\left(B\right)$.
In the above question:
$P\left(E_1\right)=\frac{2}{4}=\frac{1}{2}$

$P\left(E_2\right)=\frac{2}{4}=\frac{1}{2}$

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

Now,
$P\left(E_1{\bigcap E}_2\right)=\frac{1}{4}=P\left(E_1\right)\times P\left(E_2\right)$
$P\left(E_1{\bigcap E}_3\right)=\frac{1}{4}=P\left(E_1\right)\times P\left(E_3\right)$
$P\left(E_2{\bigcap E}_3\right)=\frac{1}{4}=P\left(E_2\right)\times P\left(E_3\right)$
Hence, $E_1,E_2$ are independent
Also, $E_1,E_3$ are independent
And $E_2,E_3$ are independent,
However, since $P(E_1\bigcap E_2{\bigcap E}_3)=\frac{1}{4}\ne P\left(E_1\right)\times P\left(E_2\right)\times P\left(E_3\right);E_1,E_2,E_3$ are not all independent.