For any two events A and B in a sample space

Options

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

Answer

A, C

Explanation

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.