Question

A bag contains four tickets marked with $112,121,211,222$ one ticket is drawn at random from the bag. Let $E_i(i=1,2,3)$ denotes the event that $i^{th}$ digit on the ticket is $2,$ then which of the following is/are true?

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

Answer 1

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

$E_3$ and $E_1$ are independent events

Total number of favourable cases is $4.$
$E_1=\{211,222\}$
$P\left(E_1\right)=\frac{1}{2}$
$E_2=\{121,222\}$
$P\left(E_2\right)=\frac{1}{2}$
$E_3=\{112,222\}$
$P\left(E_3\right)=\frac{1}{2}$
$(E_1\cap E_2)=\left\{222\right\}\Rightarrow P(E_1\cap E_2)=\frac{1}{4}$
$(E_2\cap E_3)=\left\{222\right\}\Rightarrow P(E_2\cap E_3)=\frac{1}{4}$
$(E_3\cap E_1)=\left\{222\right\}\Rightarrow P(E_3\cap E_1)=\frac{1}{4}$
$(E_1\cap E_2\cap E_3)=\left\{222\right\}\Rightarrow P(E_1\cap E_2\cap E_3)=\frac{1}{4}$

$\Rightarrow P(E_1\cap E_2)=P\left(E_1\right)P\left(E_2\right)$
$\Rightarrow P(E_2\cap E_3)=P\left(E_2\right)P\left(E_3\right)$
$\Rightarrow P(E_3\cap E_1)=P\left(E_1\right)P\left(E_3\right)$
$\Rightarrow P(E_1\cap E_2\cap E_3)\ne P\left(E_1\right)P\left(E_2\right)P\left(E_3\right)$