Let A,B, C be three mutually independent events. Consider the two statements `S_(1)and S_(2).`
`{:(S_(1):A and B nnC "are independent.",),(S_(2):A and B nnC "are independent.",):}` Then

Options

A. both `S_(1) and S_(2)` are true
B. only `S_(1)` is true
C. only `S_(2)` is true
D. neither `S_(1)` nor `S_(2)` is true

Answer

A

Explanation

We are given that
`P(AnnB)=P(A)P(B)`
`P(BnnC)=P(B)P(C)`
`P(CnnA)=P(C)P(A)`
`P(AnnBnnC)=P(A)(B)P(C)`
We have, `P(Avv(BnnC)=P(A)(B)P(C)=P(A)P(B)P(C)=P(A)P(BnnC)`
`impliesA and BnnC` are independent
Therefore, `S_(2)` is true. Also,
`P[(Ann(BuuC)]=P[(AnnB)uu(AnnB)nn(AnnC)]`
`=P(AnnB)+P(AnnC)-P(AnnBnnC)`
`=P(A)P(B)+P(A)P(C)-P(A)P(B)P(C)`
`P(A)[P(B)+P(C)-P(B)P(C)]`
`P(A)[P(B)+P(C)-P(BnnC)]`
`=P(A)P(BnnC)`