Question

Let two fair six-faced dice A and B be thrown simultaneously. If $E_1$
is the event that A shows up four, $E_2$
is the event that 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 $E3$
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.