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 Select the correct answer from above options

Answer 1

Correct Answer - D 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 .