Which of the following is (are) incorrect ?

Options

A. P(Exatly two of A, B, C occur) `leP(A cap B)+P(B cap C)+P(C cap A)`
B. `P(A cup B cup C) le P(A)+P(B)+P(C )`
C. P(Exactly one of A, B, C occur) `le P(A)+P(B)+P(C )-P(B cap C)-P(C cap A)-P(A cap B)`
D. P(A and at least one of B c, occurs)`ge P(A cap B)+P(A cap C)`

Answer

D

Explanation

We have,
P(Exactly two of A, B, C occur)
`=P(A cap B cap overline(C ))+P(A cap overline(B) cap C)+P(overline(A) cap B cap C)`
`=P(A cap B)-P(A cap B cap C)+P(A cap C)-P(A cap B cap C)+P(B cap C-P(A cap B cap C))`
`=P(A cap B)+P(B cap C)+P(A cap C)-3P(A cap B cap C) le P(A cap B)+P(B cap C)+P(A cap C)`
Also,
`P(A cup B cup C)`
`=P(A cup B)+P(C )-P{(A cup B) cap C}`
`le P(A cup B)+P(C )`.
`le P(A)+P(B)+P(C ) " " [becauseP(A cup B) le P(A)+P(B)]`
Now,
P(Exactly one of A, B, C occurs)
`=P(A cap overline(B) cap overline(C ))+P(overline(A) cap overline(B) cap C)+P(overline(A) cap B cap overline(C ))`
`=P(A cap overline(B cup C))+P(overline(A cup B)cap C)+P(B cap overline(A cup C))`
`=P(A)-P{A cap (B cap )}+P(C )-P{C cap (A cup B)}+P(B)-P{B cap (A cup C)}`
`=P(A)-P{(A cap B) cup (A cap C)+P(C )-P{(C cap A) cup (C cap B)}+P(B)-P{(B cap A) cap (B cap C)}`
`=P(A)+P(B)+P(C )=2P(A cap B)-2P(B cap C)-2P(A cap C)+3P(A cap B cap C)`
`[P(A)+P(B)+P(C )-P(A cap B)-P(A cap cap)]`
`-P[(A)+P(B)+P(C )-P(A cap C)-3P(A cap B cap C)]`
`=P(A)+P(B)+P(C )-P(A cap B)-P(B cap C)-P(A cap C)-P("Exactly two of A, B, C occur")`
`le P(A)+P(B)+P(C )-P(A cap B)-P(B cap C)-P(A cap C)`
Finally,
P(A and atleast one of B, C occurs)
`=P[A cap B cap C)]`
`P[(A cap B) cup (A cap C)]`
`=P(A cap B)+P(A cap C)-P(A cap B cap C)`
`le P(A cap B)+P(A cap C)`
So, option (d) is incorrect.