A bag contains four tickets marked with numbers `112,121,211,` and `222`. One ticket is drawn at random from the bag. Let `E_(i)(i=1,2,3)` denote the event that `ith` digit on the ticket is `2`. Then

Options

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)`.`E_(3)` are independent

Answer

A, B, C

Explanation

`(a,b,c)` Let event `E_(1)` is first digit is `'2'implies211 or 222`
`P(E_(1))=(2)/(4)=(1)/(2)`
Let event `E_(2)` is second digit is `'2'implies121, 222`
`P(E_(2))=(1)/(2)`
Let event `E_(3)` is third digit is `'2'implies222` and `112`
`P(E_(3))=(1)/(2)`
`(E_(1)nnE_(2))=222impliesP(E_(1)nnE_(2))=(1)/(4)=P(E_(1))P(E_(2))`
Similarly `E_(2)` and `E_(3)`, `E_(1)` and `E_(3)`
Also `E_(1)nnE_(2)nnE_(3) to 222`
`impliesP(E_(1)nnE_(2)nnE_(3))=(1)/(4) ne P(E_(1))P(E_(2))P(E_(3))`