Question

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.
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. `E_(1), E_(2) " and " E_(3)` are independent

Answer 1

Correct Answer - D
We find that `P(E_(1))=(6)/(36),P(E_(2))=(6)/(36)=(1)/(6)`,
`P(E_(3))=(1)/(2),P(E_(1) cap E_(2))=(1)/(36), P(E_(2) cap E_(3))=(3)/(36)=(1)/(12)`,
`P(E_(1) cap E_(3))=(3)/(36)=(1)/(12) " and " P(E_(1) cap E_(2) cap E_(3))=0`
Clearly, `P(E_(1) cap E_(2))=P(E_(1))P(E_(2)) implies E_(1), E_(2)` are independent
`P(E_(1) cap E_(3))=P(E_(1))P(E_(3)) implies E_(1), E_(3)` are independent
But, `P(E_(1) cap E_(2) cap E_(3)) ne P(E_(1))P(E_(2))P(E_(3))`
`implies E_(1),E_(2),E_(3)` are not independent
Thus, option (d) is incorrect.