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(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

Clearly , `E_(1)={(4,1),(4,2),(4,3),(44,),(4,5),(4,6)}`
`E_(2)={(1,2),(2,2),(3,2),(4,2),(5,2),(6,2)}`
and `E_(3)={(1,2),(1,4),(1,6),(2,1),(2,3),(2,5),(3,2),(3,4),(3,6),(4,1),(4,3),(4,5),(5,2),(5,4),(5,6),(6,1),(6,3),(6,5)}`
`implies P(E_(1))=(6)/(36)=(1)/(6),P(E_(2))=(6)/(36)=(1)/(6)`
and ` P_(E_(3))=(18)/(36)=(1)/(2)`
Now , `P(E_(1) cap E_(2) )=` P( getting 4 on die A and 2 on die B )
`=(1)/(36)=P(E_(1))*P(E_(2))`
`P(E_(2) cap E_(3))`=P(getting 2 on die B and sum of numbers on both dice is odd ) `=(3)/(36) =P(E_(2)) *P_(E_(3))`
`P(E_(1) cap E_(3)) `=P(getting 4 on die A and sum of numbers on both dice is odd )
`=(3)/(36)=P(E_(1))*P(E_(3))`
` and P(E_(1) cap E_(2) cap E_(3))`=P[getting 4 on die A , 2 on die B and sum of numbers is odd ] =P( impossible event )=0
Hence , `E_(1),E_(2) and E_(3)` are not independent .