Question

If A_1,A_2 and A_3 are independent events,then check whether or not
(i)A_1 and A^c_2 are independent;
(ii)A_1 and A^c_2UA^c_3 are independent..

Answer 1

(i) Recall that two events $A_i$ and $A_j$ are independent iff. $P\left(A_i\cap A_j\right)=P\left(A_i\right)P(A_j)$. We may check this:

$P\left(A_1\cap {A_2}^c\right)=P\left(A_1\cap \left(A_2\sqcup {A_2}^c\right)\right)-P\left(A_1\cap A_2\right) =P\left(A_1\right)-P\left(A_1\right)P\left(A_2\right)=P\left(A_1\right)\left(1-P\left(A_2\right)\right) =P\left(A_1\right)\left(P\left(A_2\sqcup {A_2}^c\right)-P\left(A_2\right)\right) =P\left(A_1\right)P\left({A_2}^c\right).$

(ii) We are given that set ${\left\{A_k\right\}}_{1,2,3}$ is not just pairwise independent, it is mutually independent. Hence $A_1$ and $A_2\cap A_3$ are independent. From (i), it is equivalent that $A_1$ and ${A_2}^c\cup {A_3}^c$ are independent too.

Answer:

(i) Yes (if ${\left\{A_k\right\}}_{1,2,3}$ is mutually or pairwise independent).

(ii) Yes (since ${\left\{A_k\right\}}_{1,2,3}$ is mutually independent).

References:

1 $\url{https://en.wikipedia.org/wiki/Independence_(probability_theory)#For_events}$

2 $\url{https://en.wikipedia.org/wiki/Independence_(probability_theory)#More_than_two_events}$