Let `0ltP(A)lt1, 0ltP(B)lt1 and P(AcupB)=P(A)+P(B)-P(A)P(B),` then,

Options

A. `P(A//B)=P(A)+P(B)`
B. `P(A cup B)^(c )=P(A^(c ))P(B^(c ))`
C. `P(A^(c )-B^(c ))=P(A^(c ))P(B^(c ))`
D. `P(B//A)=P(B)-P(A)`

Answer

B

Explanation

We have,
`P(A cup B)=P(A)+P(B)-P(A)P(B)`
`implies P(A)+P(B)-P(A cap B)=P(A)+P(B)-P(A)P(B)`
`implies P(A cap B)=P(A)P(B)`
`implies` A and B are independent events
`implies A^(c ) " and " B^(c )` are independent events
`implies (A^(c ) cap B^(c ))=P(A^(c ))P(B^(c ))`
`implies P(A cup B)^(c ) =P(A^(c ))P(B^(c ))`
Hence, option (b) is true.