Question 2.27: If the events A, B, and C are independent, then...

Clearly,

P(A\cup B\cup C)=P[(A^{c}\cap B^{c}\cap C^{c})^{c}]\qquad\qquad({\mathrm{by}}\,{\mathrm{DeMorgan's}}\,\mathrm{laws})

=1-[1-P(A)][1-P(B)][1-P(C)].

For any event A, P(A^{c})=1-P(A).\qquad\qquad \mathrm {basic~property~(3)}

THEOREM 6

(i) If the events A_1, A_2 are independent, then so are all three sets of events: A_{1},A_{2}^{c};A_{1}^{c},A_{2};A_{1}^{c},A_{2}^{c}.
(ii) More generally, if the events A_{1},\ldots, A_{n} are independent, then so are the events A_{1}^{\prime},\ldots ,A_{n}^{\prime} where A_{i}^\prime stands either for A_{i} or A_{i}^{c},\,i=1,\ldots,n.