Let two fari six-faced dice A and B be thrown simltaneously. 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 isodd, then which of the following statement is NOT True ?

Options

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

D

Explanation

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.