Let two fair six-faced dice A and B be thrown simultaneously. If `E_1` is the event that die A shows up four, `E_2` is the event that die B shows up two and `E_3` is the event that the sum of numbers on both dice is odd, then which of the following statements is NOT true ? (1) `E_1` and `E_2` are independent. (2) `E_2` and `E_3` are independent. (3) `E_1` and `E_3` are independent. (4) `E_1` , `E_2` and `E_3` are independent.

Options

A. `E_(2)and E_(3)` are independent
B. `E_(1)and E_(3)` are independent
C. `E_(1)and E_(2) and E_(3)` are independent
D. `E_(1)and E_(2)` are independent

Answer

C

Explanation

`E_(1):{(4,1),...,(4,6)}to6"cases"`
`E_(2):{(1,2),...,(6,2)}to6"cases"`
`E_(3):"18 cases (sum of both is odd")`
`thereforeP(E_(1))=3/6=1/6=P(E_(2))`
`P(E_(3))=18/36=1/2`
`P(E_(1)nnE_(2))=1/36`
`P(E_(2)nnE_(3))=3/36=1/12`
Similarly `P(E_(3)nnE_(1))=1/12`
`P(E_(1)nnE_(2)nnE_(3))=0`
`therefore E_(1),E_(2),E_(3)` are not independent.