For two events `A` and `B`, if `P(A)P((A)/(B))=(1)/(4)` and `P((B)/(A))=(1)/(2)`, then which of the following is not true ?

Options

A. `A` and `B` are independent
B. `P((A')/(B))=(3)/(4)`
C. `P((B')/(A'))=(1)/(2)`
D. none of these

Answer

D

Explanation

`(d)` `P((B)/(A))=(1)/(2)implies(P(BnnA))/(P(A))=(1)/(2)impliesP(BnnA)=(1)/(8)`
`P((A)/(B))=(1)/(4)implies(P(AnnB))/(P(B))=(1)/(4)impliesP(B)=(1)/(2)`
`P(AnnB)=(1)/(8)=P(A)*P(B)`
`:.` Events `A` and `B` are independent
Now, `P((A')/(B))=(P(A'nnB))/(P(B))=(P(A')P(B))/(P(B))=(3)/(4)`
and `P((B')/(A'))=(P(B'nnA'))/(P(A'))=(P(B')P(A'))/(P(A'))=(1)/(2)`