Question

For any two events A and B in a sample space A. `P((A)/(B)) ge(P(A)+P(B)-1)/(P(B)),P(B) ne 0` is always true B. `P(Acapoverset(" "-)(B))=P(A)-P(AcapB)` does not hold C. `P(AcupB)=1-P(underset(-)overset(" "-)(A))P(underset(-)overset(-)(B))`if A and B are independent D. `P(AcupB)-1-P(overset(" "-)(A))P(overset(" "-)(B))`if A and B are disjoint Select the correct answer from above options

Answer 1

Correct Answer - A::C We know that, `P((A)/(B)) = (P(A nn B))/(P(B)) = (P(A) + P(B) -P(A nn B))/(P(B))` Since, ` P(A uu B) lt 1` `rArr -P(A uu B) gt -1` `rArr P(A) + P(B) - P(A uu B) gt P(A) + P(B) -1` `rArr (P(A) + P(B) - P(A uu B))/(P(B)) gt (P(A) + P(B) -1)/(P(B))` `rArr P((A)/(B)) gt (P(A) + P(B)-1)/(P(B))` Hence, option (a) is correct. The choice (b) holds only for disjoint i.e. `P(A nn B) = 0` Finally, `P(A uu B) = P(A) + P(B) - P(A nn B)` ` = P(A) + P(B) - P(A) * P(B),` if A, B are independent ` = 1-{1-P(A)} {1-P(B)} = 1-P (bar(A)) * P(bar(B))` Hence, option (c) is correct, but option (d) is not correct.