For any two events A and B, which one of the following holds?

Options

A. `P(A cap B) le P(A) le P(A cup B) le P(A)+P(B)`
B. `P(A cup B) le P(A) le P(A cap B) le P(A)+P(B)`
C. `P(A cup B) le P(B) le P(A cap B) le P(A)+P(B)`
D. `P(A cap B) le P(B) le P(A)+P(B) le P(A cup B)`

Answer

A

Explanation

Clearly, `A cap B subseteq A`
`Rightarrow P(A cup B) le P(A).....(i)`
`A subseteq A cup B`
`Rightarrow P(A) (A cup B)....(ii)`
We know that, `P(A cup B)=P(A)+P(B)-P(A cap B)`
`Rightarrow P(A cup B) le P(A)+P(B).....(iii)`
From eqs. (i), (ii) and (iii), we get
`P(A cap B) le P(A) le P(A cup B) le P(A)+P(B)`